random holonomy, large unitary matrices · after ambar sengupta had given the rst one. since then,...

Habilitation a diriger les recherches

Specialite : Mathematiques

Random holonomy,

large unitary matrices

Thierry LEVY

Rapporteurs : M. Bruce DRIVERM. David ELWORTHYM. Wendelin WERNER

Soutenue le 20 novembre 2009 devant le jury compos de :

M. Philippe BIANEM. Philippe BOUGEROLM. David ELWORTHYMme Alice GUIONNETM. Yves LE JANM. Wendelin WERNER

Introduction

This text is a presentation of my research in mathematics since the beginning of my PhDthesis. The structure of this text has been strongly influenced by the nature of this workand I shall first explain how.

There is one problem which has been constantly at the centre of my circle of interests,namely the construction and the study of the two-dimensional Yang-Mills measure. Thismeasure is a mathematically tractable simplification of one of the fundamental ingredientsof the formulation with Feynman integrals of gauge theories, which have been, since almostfour decades, the best tool available to physicists for the description, at infinitesimal scale,of the ultimate constituents of matter and their interactions.

My PhD thesis was devoted to giving a second rigorous construction of this measure,after Ambar Sengupta had given the first one. Since then, I have used this constructionto study several aspects of the measure and at the same time I have tried to improve itand to generalise it. This has led me to define a class of stochastic processes, which Icall Markovian holonomy fields, which contains the Yang-Mills measure pretty much inthe same way as the class of Levy processes contains the Brownian motion. In additionto this, during the last few years, the study of the so-called large N limit of the Yang-Mills measure has driven my attention towards questions about large unitary matricesthe significance of which is, to a large extent, independent of my original motivation forconsidering them.

The thread running through my research to this date is therefore that which leadsfrom the study of the Yang-Mills measure to the definition of Markovian holonomy fields.Nevertheless, neither the Yang-Mills measure nor these Markovian holonomy fields areclassical or simply well-known objects. Thus, instead of writing a list of results with ashort description of their context, I have preferred to conceive a large part of this text asan introduction to the theory of random holonomy fields.

The first two chapters of these notes are precisely that. They can be seen as anoverview of the monograph [L7]1 or, even better, as a guide to reading it. The first chapteris probabilistic in nature and the second more geometric. Indeed, one of the peculiaritiesof the work that is described here is that it uses both the language of probability andthat of differential geometry. Among many other options, I have chosen to write this textfor a reader familiar with probability and not hostile to differential geometry. The mostimportant geometric objects are thus described in detail, presumably in much more detail

1This monograph will be published in the course of the year.

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than an expert would like. Moreover short reviews of various subjects are included in theform of inserts, so that the reader can easily refer to the ones which are useful to himand, just as easily, ignore those which are not.

The third chapter presents my work, partially in collaboration with Florent Benaych-Georges and Mylene Maıda, on large random unitary matrices. In this field, which islikely to be more familiar to many potential readers, I have reckoned that a shorterpresentation of the context would be sufficient. Nevertheless, in this chapter too, theobjects are carefully introduced and the progression of the text is punctuated by inserts.

Finally, as an appendix to these notes, I have added a text which I would have lovedto have at hand when I was starting my own research on the Yang-Mills measure. In thistext, I sketch a path, albeit superficial, from Maxwell’s equations to gauge theories, withan emphasis on the reason why it is natural to introduce principal bundles in the quantummechanical formulation of electromagnetism. This text is a patchwork of fragments whichI have gathered here and there, and its only originality may be the fact that it is writtenby a mathematician and for mathematicians.

Presentation of the results

I will now describe in greater detail the subject of my research, and in the first place howit is issued from theoretical physics.

The mathematical difficulties of gauge theories

The Yang-Mills measure bears the names of the two physicists Chen Ning Yang and RobertL. Mills who, in 1954, considered for the first time the possibility of gauge fields takingtheir values in non-Abelian Lie algebras in order to account for certain effects of the stronginteraction [YM54]. In doing so, they opened the way to non-Abelian gauge theories,which have provided physicists with the framework needed to progressively build the so-called standard model which has been, since the beginning of the 1970’s the best availabletheory for the nature and interaction of the fundamental constituents of matter, witha conceptually unified description of the electromagnetic, weak, and strong interactions[CG07].

In the standard model, the interactions are conveyed by gauge bosons, like photons,which convey the electromagnetic interaction 2 These bosons are represented by fluctua-tions of a gauge field, whose mathematical nature is that of a connection on a principalbundle. The theory of connections was precisely formalised by Charles Ehresmann in the1950’s [Ehr51], for internal reasons in mathematics. It is said that Chen Ning Yang andShiing-Shen Chern, in a discussion around 1970, both wondered why the other found sonatural to consider such objects in their own field.

From the physical point of view, a gauge theory is characterised by its Lagrangian,which expresses how the various fields under consideration interact. The Yang-Mills

2In order to imagine how two electrons can repulse each other by exchanging photons, one can think of twoice- skaters moving forward on parallel tracks and throwing a very heavy ball to each other, one after the other,thus each time inducing their trajectories to diverge slightly.

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Lagrangian, in a space without matter and with only gauge bosons, is given by theexpression L(A) = Tr(F ∧ ∗F ), where F is the strength field of the gauge field A, that isthe curvature of the connection A. This Lagrangian being given, one defines the Yang-Mills action as the integral

S(A) =

∫R4

L(A),

so that S is a real-valued functional on the space, traditionally denoted by A, of allconnections. In accordance with Feynman’s formulation of quantum mechanics, one canthen compute the probability that a certain physical event occurs by computing the squareof the modulus of an integral of the form∫

BΦ(A)e

i~S(A) dA, (1)

where B ⊂ A is a certain subset of connections, Φ : A → C is a functional, and dA areference measure on A.

Expressions like (1) have no mathematical meaning. The most obvious problem isthe fact that the space A of connections is not locally compact 3 and does not carry anyσ-finite Radon measure invariant by translations, although that is what dA should be,according to the use that physicists make of it. Nevertheless, the incredible precision of thepredictions which physicists extract from such integrals, and more generally from quantumfield theories, indicate that it is possible and important to uncover the mathematicalnature of the objects involved in these theories.

Generally speaking, finding a rigorous mathematical framework for the formulation ofthe standard model is still a widely open problem4, but in the case of a two-dimensionalspace-time and in a Euclidean situation (that is, replacing in (1) the i of the exponentialwith a −1), it is possible to give a rigorous meaning to integrals of the form (1) whichis consistent with their physical meaning. One can, in this way, give a meaning to theexpression

µ(dA) =1

Ze−

12S(A) dA, (2)

which is the most common heuristic description of the Yang-Mills measure, as a probabilitymeasure on the space A of all connections5.

In the context which we will consider here, with a two-dimensional space-time, onecan, as a first approximation, think of a connection as a differential 1-form on a surface.A typical functional of a connection is thus its integral along a path, or along a loop.In order to be less approximate, it is better to think of a connection as a multiplicative

3One could think that the gauge symmetry of the theory solves this problem, since it suffices to work onthe quotient of A under the action of the gauge group. But, contrarily to a wrong idea which I have heardexpressed several times, the quotient of the space of connections by the action of the gauge group is not locallycompact nor finite-dimensional in any sense. However, the moduli space of flat connections, quotiented by gaugetransformations on a principal bundle over a reasonable space (for example a space whose fundamental group isfinitely generated) is a finite-dimensional object.

4It is one of the six remaining problems for the resolution of which the Clay institute offers one million dollars.See www.claymath.org/millenium/Yang-Mills Theory.

5For more details on the discussion so far, one can consult Section 2.3 and in particular Definition 2.3.1.

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version of a differential form. A group G is given, called the structure group, which couldfor example be SU(2), and the connection determines, by integration along each path, anelement of this group. This element is called the holonomy of the connection along thepath. Of course, the holonomy along the concatenation of two paths is the product of theholonomies along each path.

It happens that only holonomies along loops carry meaningful information in gaugetheories, because of the symmetry of these theories which is precisely called gauge sym-metry. The structure group G being fixed, one can thus think of a connection as anobject which to each loop traced on the surface associates an element of G. This can bewritten in a concise (but slightly incorrect) way as follows: if M is a surface and L(M)is the set of loops on M , and if G is a compact Lie group, then we have the inclusion6

A ⊂ fonctions de L(M) dans G.From this point of view, defining a probability measure on the space of connections

amounts to defining a probability on the set of functions from the set of loops on oursurface and with values in the structure group G. This idea is absolutely fundamentalfor all that follows, as we are going to describe objects of the latter type: not randomdifferential forms, but stochastic processes indexed by loops traced on surfaces and withvalues in a compact Lie group.

In order to help the imagination, let us say that the processes which we are going toconsider are Levy7 processes indexed by loops and with values in a group. If the surfacewhich we consider is for example a plane, and if for each t ≥ 0 we consider the randomvariable associated with a circle which goes once positively around a disk of area t (insuch a way that the interiors of these circles increase with t, see Proposition 1.1.3) thenwe obtain a stochastic process indexed by R+ with values in G, whose multiplicativeincrements are independent and stationary. If the surface is a sphere, then our familyof loops is going, after a time equal to the total area of the sphere, to shrink back to aconstant loop after “going around the sphere”. In this case, we will rather obtain a bridgeof a Levy process with values in G. Surfaces with different topologies will lead to variantsof this bridge with values in G.

Part of the problems which we are going to consider are thus related to these processes,their definition, through discretisation, their characterisation, their classification, theirgeometrical meaning, and the study of their asymptotic behaviour in certain limits. Weare in particular going to consider the limit in which the total area of the surface on whichone works is multiplied by a constant which tends to zero (the semiclassical limit of thetheory), for which we present a large deviations principle, proved in collaboration withJames R. Norris. The study of another limit, when the structure group is the unitary groupU(N) and N tends to infinity, will not be discussed directly here, but is the motivationfor tackling some questions related to large random unitary matrices which are presentedbelow and will be exposed in detail in the last chapter. Some of these questions have beensolved in collaboration with Florent Benaych-Georges and Mylene Maıda.

6For a definition of the gauge symmetry and a correct statement of this inclusion, see 2.2.4 and what precedesit. For a discussion of the relation between gauge symmetry and the choice of a gauge in electromagnetism, see,in the appendix, Sections A.3 and A.4.

7This is Paul Levy, 1886-1971.

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The construction of the Yang-Mills measure

After the seminal work of Yang and Mills in 1954 [YM54], one traditionally refers to apaper of Alexander A. Migdal written in 1975 [Mig75] as the first notable step towards amathematically rigorous description of the discrete Yang-Mills measure in two dimensions.In this paper, Migdal identifies indeed, maybe for the first time, the fundamental roleplayed by the heat kernel on the structure group. This kernel is not named, but writtenas the sum of its Fourier series.

In the 1980’s, Sergio Albeverio, Raphael Høegh-Krohn and Helge Holden examined,in an essentially discrete framework, the idea of a random measure on a surface taking itsvalues in a Lie group [AHKH86, AHKH88]. In particular, they understood the importancefor this problem of convolution semi-groups. The Markovian holonomy fields which arepresented here are, as we shall see, in correspondence with convolution semigroups ofmeasures on the structure group and this can be seen as a continuation of the ideas ofthese authors.

The problem of the construction of the continuous Yang-Mills measure in two dimen-sions was first considered in the 1980’s by Leonard Gross [Gro85, Gro88], then studied byBruce Driver [Dri89, Dri91] in the case where the surface is a plane, and was finally re-solved by Ambar Sengupta on an arbitrary compact surface, around 1994 [Sen92, Sen97a].The approach of Gross, Driver and Sengupta is based on the observation of the fact that,provided one works in an appropriate gauge, a connection taken under the Yang-Millsmeasure can, according to (2), be understood as a connection whose curvature has thedistribution of a white noise8, possibly conditioned in a way which depends on the topol-ogy of the surface on which one works. It is thus by giving himself a white noise and bymaking sense of a singular conditioning of this noise that Ambar Sengupta has achievedthe construction of a sort of Brownian motion indexed by the loops traced on a compactsurface and with values in a connected compact Lie group9

In the early 1990’s, Edward Witten published two papers on two-dimensional Yang-Mills theory [Wit91, Wit92], which of course raised great interest. In one of these papers[Wit91], Witten describes the main features of the discrete Yang-Mills theory and uses itto study the geometry of the moduli space of flat connections on a Riemann surface, andmore specifically to compute its symplectic volume. The measure of symplectic volume ofthe moduli space of flat connections can indeed be understood as the semi-classical limitof the Yang-Mills measure, that is, its limit as the total area of the space-time tends tozero. I have not studied this aspect of the measure and I will not discuss it in this text10

Nevertheless, the large deviation principle James Norris and I have established, and towhich I will come back in a moment, studies a limit, but of another nature, as the totalarea of the space-time tends to zero.

Around the same time, other more or less partial approaches of the problem of the

8This white noise takes its values in the Lie algebra of the structure group.9This construction is analogous to the construction of the usual Brownian motion on [0, 1] using a white noise

W on [0, 1], that is, an isometry of L2([0, 1], dt) into a linear space of centred Gaussian variables. One sets, forall t ∈ [0, 1], Bt = W (1[0,t]). Conditioning the noise by the equality W (1[0,1]) = 0, one gets a Brownian bridge.

10For more detail on this aspect of the measure, the reader is kindly invited to consult the works of MichaelAtiyah and Raoul Bott [AB83], William Goldman [Gol84], Robin Forman [For93], Kefeng Liu [Liu96, Liu97],Ambar Sengupta and Christopher King [Sen97b, KS94].

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construction of the Yang-Mills measure were proposed which, although not all math-ematically completely satisfactory, have shed light on various aspects of this measure.I think in particular to the work of Dana Fine [Fin90, Fin91], Claas Becker [Bec95],Doug Pickrell [Pic96], John Baez and Stephen Sawin [BS97, BS98], Abhay Ashtekar etal. [AI92, AL94, AL95, AMMa94].

In my first research [L1, L2], I gave another construction of the Yang-Mills measure,and this constituted my PhD thesis, under the direction of Yves Le Jan. My approachwas to rigorously ground the discrete theory described by Witten, and then to prove thatit is possible to pass it to the continuous limit11. More precisely, in order to construct aprocess of random holonomy indexed by loops traces on a surface, one describes its finite-dimensional marginals and then applies a version of Kolmogorov’s theorem. However, thefact that these processes are indexed by paths creates specific problems, as for example thefact that it is not possible to describe in a simple way all its marginals. Approximationsare thus required, the quality of which determines the class of loops along which one willeventually be able to define the random holonomy. In [L2], I was able to define a processindexed by piecewise smooth paths and associated, in the sense alluded to a few pagesago, with the Brownian motion on a connected compact Lie group.

In a later work [L4], I generalised the discrete part of the construction in order to beable to include some topological refinements12. This allowed James Norris and I, duringour joint work in Cambridge [LN5], to prove a large deviation principle. This result is thefirst, and, to this day, the unique rigorous result which relates the Yang-Mills measureto the Yang- Mills energy functional. In this work, it was necessary to consider genuineprincipal bundles and genuine connections on these bundles, and to confront them withthe functions of loops which were supposed to be their holonomy13

Besides, following Bergfinnur Durhuus [Dur80] and d’Ambar Sengupta [Sen94], I stud-ied an algebraic aspect of the discrete Yang-Mills theory and improved a density resultwhich is of some theoretical importance in order to justify the almost exclusive use thatphysicists make of Wilson loops as observables in gauge theories [L3]. More precisely, it isnot obvious that the class of observables that is usually considered by physicists, namelythe algebra generated by Wilson loops, which are the traces of the holonomies alongloops, is complete, in the sense that it separates the points of the configuration space.The cases where the structure group is unitary or odd orthogonal, or a product of suchgroups, had been solved positively by Durhuus and then Sengupta. Using some classicalinvariant theory and another natural class of observables, the spin networks, invented in

11To continue the previous analogy, this approach corresponds to the construction of the Brownian motion asa limit of random walks.

12The problem was to construct one measure for each class of isomorphism of principal bundle on the surfaceconsidered, in the case where there exist non-trivial bundles. The classical discrete theory sees a bundle throughits restrictions to graphs, that is, 1-dimensional objects, and, as long as the structure group is connected, it cannotdetect the difference between two non-isomorphic bundles. In order to solve this problem, I replaced the usualconfiguration space of the discrete theory by a larger space, which outside a singular set is a covering of it. Takingthe continuous limit, I obtained measures associated to specific bundles and this allowed me to disintegrate themeasure constructed in [L2]. I also proved that the holonomy processes associated to non-isomorphic bundleshave mutually singular distributions.

13In particular, we were led to solve minimisation problems under constraints and to explicitly construct Lip-schitz continuous connections whose holonomy was prescribed along the edges of a graph and whose action wasminimal under this constraint.

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the 1970’s by Roger Penrose and introduced in the 1990’s by John Baez in gauge theories(see [Bae96]), I have proved that the algebra generated by Wilson loops is complete whenthe structure group is any product of unitary, orthogonal and symplectic groups. Thecase of spin groups is still open.

Despite the existence of two rigorous approaches, that of Ambar Sengupta and mine,for the construction of the Yang-Mills measure, and despite the generalisations that I hadadded to mine, it seemed to me that several points needed to be clarified and that theconstruction had not at all reached its final form.

Large unitary matrices

Around 2005, I started to study an aspect of the Yang-Mills measure that I had ignoreduntil then, namely its behaviour when the gauge group is the unitary group U(N) and Ntends to infinity. This limit is usually simply called the large N limit of the theory. IsadoreSinger has published a short paper [Sin95] on this subject, in which he opens vast perspec-tives where essentially everything is still to be done. Michael Anshelevich developed andexplained this paper as far as possible [Ans97]. Nevertheless, the literature in this fieldis still essentially the physical literature [GG95, GM95, GT93b, GT93a, Kaz81, KK80].The first questions which arise in this direction of research can in fact be formulated quiteindependently from the Yang-Mills measure and are related to large unitary matricestaken under the heat kernel measure. These random matrices have been studied mainlyby Philippe Biane [Bia97], Feng Xu [Xu97], Steve Zelditch [Zel04]. More recently, AmbarSengupta [Sen08] has clarified the work of F. Xu.

In [L6], I established a relation between the Brownian motion on the unitary groupand a very simple random walk on the symmetric group, which can be understood as aconsequence of the Schur-Weyl duality or of the Ito formula. Understanding this relationhas allowed me to give a combinatorial expression of the moments of the distribution ofthe eigenvalues of a unitary matrix of size N taken under the heat kernel measure. Thisexpression is a convergent power expansion in non-positive even powers of N , which canbe interpreted rigorously as an expansion according to the genus of a certain randomramified covering of a disk. This interpretation gives a rigorous proof of the simplest ofthe wonderful formulas given by David Gross and Washington Taylor in [GT93b, GT93a].The expression that I have given allows one also to recover the moments of the limitingdistribution of the eigenvalues, already computed by P. Biane, and results of asymptoticfreeness (in the sense of free probability).

The familiarity with the unitary Brownian motion gained during this work allowedme, in collaboration with Florent Benaych-Georges, to construct in [LBG8] a family ofstructures of dependence (or independence) in a non-commutative probability space whichinterpolate between independence and freeness. It was known that, if A and B are twolarge diagonal real matrices whose eigenvalues are approximately distributed according totwo probability distributions µ and ν, and if U is a permutation matrix chosen uniformlyat random (resp. a unitary matrix chosen under the Haar measure), then the eigenvaluesof A + UBU−1 are distributed according to the measure µ ∗ ν, the classical convolutionof µ and ν (resp. according to µ ν, the free convolution of µ and ν). Giving to U thedistribution of a Brownian motion at time t on the unitary group whose distribution at

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time 0 is the uniform measure on the group of permutation matrices, we have definedan operation of convolution ∗t for all non-negative real t, which for t = 0 is the classicalconvolution and for t tending to infinity tends to the free convolution. In fact, we definedthe structure of dependence between two sub-algebras of a non- commutative probabilityspace which underlies this convolution. Our initial hope was to identify cumulants asso-ciated with this t-free convolution, that is, universal multilinear forms the cancellationof which would characterise the t-freeness of some of their arguments. We thought thatthey might interpolate between classical cumulants, which are intimately connected to thecombinatorics of the partitions of a set, and free cumulants, related to the non-crossingpartitions of a set endowed with a cyclic order. This hope has been turned down [*] andwe have in fact proved that there are no t-free cumulants.

In a work in collaboration with Mylene Maıda [LM9], we established a central limittheorem for functions on the unitary group of the form U 7→ trf(U), where f is a suffi-ciently regular function on the unit circle of the complex plane and tr is the normalisedtrace. The function f being given, the expression trf(U) has a meaning for U elementof U(N) for all N . The results of P. Biane [Bia97] describe the limit as N tends to in-finity of this function when U has the distribution of a Brownian motion at time t. Thislimit is a deterministic and almost-sure limit. We have studied the fluctuations associatedwith this convergence, proved that they are Gaussian, and determined the quadratic formon the space of regular functions which gives the covariance of these fluctuations. Thiscovariance is naturally expressed in the framework of free probability, in terms of thefree unitary Brownian motion. We have been able to relate our result to that obtainedby Persi Diaconis and Steven Evans [DE01] in the case of unitary matrices taken underthe Haar measure, by showing that the covariance which they had identified, namely thescalar product in the Sobolev space H

12 , is the limit of our covariance as time tends to

infinity.

Markovian holonomy fields

In parallel to these works on large unitary matrices, I realised during the summer 2006that it was possible to prove a certain property of continuity for the Yang-Mills measure,and that this allowed me to improve the apparently secondary and fairly technical pointthat is the class of loops with which one is working. In this respect, I was then able towork in a much more natural framework than ever before. The point was to extend thedefinition of random holonomy to the set of all loops with finite length, which is probablythe largest class which one can deal with using only the techniques I have at my disposal[*]. The possibility of this extension has pushed me to reconsider the construction fromthe start and, progressively, I have been led to define Markovian holonomy fields [L7],thus completely generalising the construction that I had given in my PhD thesis. TheMarkov property of the random holonomy fields has become a central part of their verydefinition, which is a mixture of the definition of well-behaved classical Markov processesand topological quantum field theories. The class of objects which I have thus definedcontains the Yang-Mills measure, which is its prototype, essentially as the class of Levyprocesses contains the Brownian motion.

One of the important aspects of the greater generality brought by Markovian holonomy

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fields is the possibility to consider non-connected structure groups and in particular finitegroups. In the case of the symmetric group, a work by Alessandro d’Adda and PaoloProvero [DP04] showed a link between a theory analogous to Yang- Mills theory and amodel of random ramified coverings. I have obtained a similar result in the general case,which allows one to realise explicitly a large class of Markovian holonomy fields with valuesin a finite group as monodromies of a random ramified covering14 whose ramification locusis essentially a Poisson point process.

Localisation of the main statements

To state the main results presented in these notes with sufficient precision would neces-sitate long introductions which, in fact, constitute most of the text to follow. I will thussimply indicate where in the text the main statements are located.

The main statements of the first chapter are:

• the definition of Markovian holonomy fields (Definition 1.3.3, p.37),

• a theorem of existence for these fields (Theorem 1.3.7, p.40),

• a theorem of extension which is the key for passing from the discrete theory to acontinuous one (Theorem 1.2.17, p.35),

• a partial classification result of these fields (Theorem 1.3.6, p.39),

• the construction of the Yang-Mills fields starting from a white noise (Theorem 1.3.9,p.42),

• a density result for Wilson loops in the algebra of observables in a discrete gaugetheory, for a certain class of structure groups (Theorem 1.2.11, p.31).

The main statements of the second chapter are:

• the geometric realisation of Markovian holonomy fields with values in finite groupsby random ramified coverings (Theorem 2.1.5, p.51),

• the disintegration of the Yang-Mills field according to the possible isomorphismclasses of principal G-bundles over a surface (Theorem 2.4.6, p.64),

• two large deviation results for the Yang-Mills measure as the total area of the surfacetends to zero, proved in collaboration with James Norris (Theorems 2.4.8 and 2.4.9,p.65).

The main statements of the third chapter are:

• a theorem which expresses in combinatorial terms the moments of a unitary matrixtaken under the heat kernel measure (Theorem 3.1.4, p.73),

14Instead of a ramified covering, one should rather consider a ramified principal bundle, that is, a ramifiedcovering on the regular fibres of which the structure group acts simply transitively.

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• a formula due to Gross and Taylor, which I have been able to prove thanks to theprevious result (Theorem 3.1.13, p.79).Then, in collaboration with Mylene Maıda:

• a central limit theorem for the repartition of the eigenvalues of a unitary matrixunder the heat kernel measure (Theorem 3.2.13, p.84),

• the convergence, as time tends to infinity, of the covariance which appears in theprevious theorem to that identified by Persi Diaconis and Steven Evans in the caseof uniformly distributed unitary matrices [DE01] (Theorem 3.2.14, p.84).Finally, in collaboration with Florent Benaych-Georges:

• the definition of the t-free convolution of two probability measures on R (Theorem3.3.1, p.86),

• the definition of t-freeness of two sub-algebras of a non-commutative probabilityspace, interpolating between the notions of freeness and independence (Definition3.3.5, p.89),

• a statement which encapsulates the differential system allowing one to compute thet-free convolution of certain distributions (Proposition 3.3.7, p.90),

• a result of non-existence of cumulants for the notions of t-freeness (Theorem 3.3.14,p.93).

Chapter 1

Markovian holonomy fields[L2, L3, L7]

1.1 Introduction

The aim of this introduction is to provide us with an overview of Markovian holonomyfields, before we take the subject from the beginning with an order of exposition closeto the logical order. We are going to study a simple and fundamental example whichwill give us a sense of the kind of Markov property which is involved here. We willthen give a temporary definition of these holonomy fields, and discuss their “transitionkernels”, whose properties are intimately related to the surgery of surfaces and can benicely drawn. We will finish this section by stating a result of existence and classification,which is one of the main achievements of our work on these processes.

Virtually all that is said in this introduction will be repeated in more detail later inthis chapter.

1.1.1 The Poisson process modulo n indexed by loops

Consider the Euclidean plane R2. Let us temporarily call loop a continuous mappingl : [0, 1] → R2 such that l(0) = l(1) and whose range is Lebesgue negligible. Let l be aloop and let z be a point of R2 identified with C, not located on the range of l. The indexof l with respect to z is the integer nl(z) defined by

nl(z) =1

2iπ

∫l

dw

w − z , (1.1)

where l is any piecewise C1 loop whose uniform distance to l is smaller than the distanceof z to the range of l, so that l and l are homotopic in R2 \ z. Since we are assumingthat the range of l is negligible, the function nl is defined and finite almost everywhere onR2, and has compact support.

Let n ≥ 2 be an integer. We are going to associate to each loop l a random variablewith values in Z/nZ. For this, we are going to add modulo n the indices of l with respectto the points of one or more Poisson point processes.

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14 CHAPTER 1. MARKOVIAN HOLONOMY FIELDS [L2, L3, L7]

Definition 1.1.1 Let λ1, . . . , λn−1 be non- negative reals. Let Π1, . . . ,Πn−1 be indepen-dent Poisson point processes on R2 such that for all k ∈ 1, . . . , n − 1, the intensity ofΠk is λk times the Lebesgue measure. To each loop l we associate the random variable

Nl =n−1∑k=1

k

∫R2

nl(z) dΠk(z) (mod n). (1.2)

Figure 1.1: Take n = 4. The processes Π1, Π2 and Π3 are respectively represented by black squares,blue crosses and red disks. So, the contribution of Π1 is 2, that of Π2 is 6 and that of Π3, 6. Hence, inthis example, Nl = 2 (mod 4).

Let us temporarily denote by L(R2) the set of loops on R2. We have just defineda stochastic process indexed by L(R2) and taking its values in Z/nZ. Let us state afew properties of this process. Given a loop l, we denote by l−1 the same loop tracedbackwards. Also, given two loops l1 and l2 based at the same point, we denote theirconcatenation by l1l2.

Proposition 1.1.2 The process (Nl)l∈L(R2) is multiplicative. This means that for all loopsl, l1, l2 ∈ L(R2) such that l1 and l2 are based at the same point, we have

Nl−1 = −Nl,

Nl1l2 = Nl1 +Nl2 .

Let us now consider a very simple one-parameter family of loops. For each t ≥ 0, letus denote by lt a loop which bounds once and positively a disk of area equal to t, thesedisks being chosen in such a way that for all s ≤ t, the disk of area t contains the diskof area s. Then (Nlt)t≥0 is an ordinary stochastic process, indexed by R+, with values inZ/nZ.

Proposition 1.1.3 The process (Nlt)t≥0 is a Markov chain on Z/nZ. Its generator(Qa,b)a,b∈Z/nZ is given, for all a 6= b, by Qa,b = λc, where c is the unique integer in1, . . . , n− 1 such that b = a+ c (mod n).

The increments of this process are independent and stationary : it is a Levy process(that is, a continuous time random walk), whose jump measure, which is a positive measureon (Z/nZ) \ 0, gives the mass λk to the class of k for all k ∈ 1, . . . , n− 1.

1.1. INTRODUCTION 15

Figure 1.2: With the same convention as in the previous picture, the successive values of Nlt for thefour values of lt depicted here are 0, 3, 1 and 0 modulo 4.

We say that the Levy process described in the proposition above is associated with theprocess (Nl)l∈L(R2). Its distribution does not depend on the way in which we have chosenthe loops (lt)t≥0. From the Markov property of this process follows a Markov property ofthe process N , which can be expressed as follows. If J is a loop which is also a Jordancurve, we will denote respectively by int(J) and ext(J) the closures of the bounded andunbounded connected components of the complement in R2 of the range of J .

Proposition 1.1.4 Let J be a Jordan curve. Then the two σ-fields σ (Nl : l ⊂ int(J))and σ (Nl : l ⊂ ext(J)) are independent given NJ .

More generally, let J, J1, . . . , Jn be pairwise disjoint Jordan curves such that the inte-riors of J1, . . . , Jn are pairwise disjoint and contained in the interior of J . Define

M1 = int(J) ∩ ext(J1) ∩ . . . ∩ ext(Jn) and M2 = ext(J) ∪ int(J1) ∪ . . . ∪ int(Jn).

Then the two σ-fields σ (Nl : l ⊂M1) and σ (Nl : l ⊂M2) are independent conditional onσ(NJ , NJ1 , . . . , NJn).

J J

J1

J2

J3

J4

Figure 1.3: Left : loops located in the exterior of J may wind around the interior of J . Right : thedomain M2 is shaded.


1.1.2 A first definition of Markovian holonomy fields

The proposition 1.1.4 expresses a Markov property where the notions of past and future arereplaced by the notions of interior and exterior, with respect to one or more curves whichsplit the plane into several connected components. Even applying this Markov propertyon such a simple domain as R2, we find ourselves dealing with more complicated domainssuch as the domain M1 depicted (in white) in the right part of Figure 1.3, homeomorphicto a sphere with five holes.

Along with the cutting operation which has for instance lead us from the plane to thefive- holed sphere, gluing operations are also natural. If for example we start from Poissonprocesses indexed by loops traced on two disjoint disks, both conditioned by the fact thatthey have a certain value x ∈ Z/nZ on the boundary of their respective disk, then it isnatural to glue these disks in order to define the Poisson process indexed by loops tracedon a sphere, or at least the conditional version of this process determined by the eventthat its value along a certain equator is x.

In order to formulate the Markov property in full generality, one should thus notrestrict oneself to the plane R2 nor to a disk, but consider a class of domains which isstable under the operations of cutting along Jordan curves and gluing along componentsof the boundary. Such a class is easy to identify : as soon as one starts with disks andthree-holed spheres, one can, by gluings, construct any compact surface. It is thus theclass of compact surfaces which constitutes the natural class of domains on which theloops which index our processes will be traced.

INSERT 1 – Topology of surfaces [Mas77, Moi77]

A compact surface, or simply a surface, is a compact smooth 2-dimensional real manifold which may or maynot be orientable and may have a boundary. As a topological space, it is a Hausdorff compact space of whicheach point admits a neighbourhood homeomorphic to R2 or to R × R+. The differentiable structure carriesno information which is not already contained in the topological structure : two surfaces are diffeomorphic ifand only if they are homeomorphic.

The boundary of a surface, if not empty, is a finite disjoint union of circles. A surface whose boundary isempty is called a closed surfaces. To each surface one can associate a closed surface by gluing a disk alongeach connected component of the boundary. Two connected surfaces with the same number of connectedcomponents of boundary are diffeomorphic if and only if the associated closed surfaces are diffeomorphic.Finally, connected closed surfaces are classified up to diffeomorphism by their orientability and their genus,which can be any non-negative integer for an orientable surface, and any positive integer for a non-orientablesurface. When one is working at the same time with orientable and non-orientable surfaces, it is moreconvenient to consider their reduced genus, equal to the genus for non-orientable surfaces and twice the genusfor orientable ones. The main advantage of this reduced genus is that it is additive with respect to theoperation of connected sum.

The closed surface of genus 0 is the sphere, the closed oriented surface of genus g (and reduced genus 2g)is the connected sum of g tori, and the non-orientable surface of genus g is the connected sum of g projectiveplanes. The projective plane can be obtained from the unit sphere of R3 by identifying all pairs of antipodalpoints.

Unlike in the 1-dimensional case, an orientation of an orientable surface does not allow one to distin-guish among the connected components of its boundary between incoming and outcoming components. Infact, any permutation of the boundary components of a surface, even an oriented one, can be realised by adiffeomorphism, actually by an orientation-preserving diffeomorphism if the surface is oriented.


Just as the Poisson process described in the previous section was taking its valuesin Z/nZ, Markovian holonomy fields will take their values in a group, which in generalwill not be Abelian. It will rather be a compact Lie group, that is to say, a compactmatrix group. We will also insist that holonomy fields satisfy a multiplicativity propertyanalogous to that expressed by Proposition 1.1.2, namely the following (see also Definition1.2.1 below) : if G is a group, M a surface and m a point of M , a family (Hl)l∈Lm(M) ofrandom variables indexed by the set of loops on M based at m is multiplicative if

∀l1, l2 ∈ Lm(M) , Hl−11

= H−1l1

and Hl1l2 = Hl2Hl1 almost surely.

The reason why the order of l1 and l2 is reversed in the last equality will be explained inSection 2.2.2.

Another important aspect of the Poisson process indexed by loops is the role playedby the Lebesgue measure on R2 in its construction. It was indeed, up to multiplicativeconstants, the intensity of the Poisson point processes Π1, . . . ,Πn−1. In the general case,we need to endow the surfaces which we consider with a measure of area, which as we shallsee plays a role analogous to a measure of lengths in the usual case of Markov processesindexed by intervals of time.

We can now give a temporary and loose definition of Markovian holonomy fields.

Definition 1.1.5 (Markovian holonomy fields : temporary definition) Let G bea compact Lie group. A bidimensional G-valued Markovian holonomy field is a collectionof G-valued stochastic processes, one for each compact surface endowed with a measureof area and a choice of boundary conditions along some of its boundary components. Foreach such surface, the process is indexed by the set of loops traced on the surface and itis multiplicative. Finally, the collection of all these processes behaves with respect to thesurgery of surfaces (cutting along Jordan curves and gluing along boundary components)in a way which is governed by the Markov property.

1.1.3 Transition kernels and surgery of surfaces

In order to understand the role of the boundary conditions which are mentioned in thedefinition above, we are going to make an analogy with the more familiar case of Markovprocesses indexed by intervals of time. The data of such a process, taking its values ina state space X , allows one to associate to each positive real t and each initial conditionx ∈ X , a probability measure Pt(x, dy) on X . The transitions of this process occuralong intervals, which can be though of, albeit pedantically, as topological transitions (orcobordisms) between pairs of points. Similarly, the transitions of a Markovian holonomyfield occur along surfaces seen as cobordisms between families of circles (see for exampleFigure 1.4 below).

Nevertheless, according to the remark made at the end of Insert 1, and contrarily towhat happens on intervals, the boundary of a surface, even an oriented one, does not splitcanonically into incoming and outgoing connected components. Besides, a surface mayhave an arbitrary non-negative number of boundary components. Looking at a particular


surface as realising a transition between certain connected components of its boundaryand the others is thus the result of an arbitrary decision. The analytic counterpart of thishigher symmetry between connected components of the boundary as between the initialand final point of an interval is the necessity to abandon the distinction which makes thetransition kernel Pt(x, dy) of a usual Markov process a function in x and a measure in y. Inthe usual case, this distinction can be overlooked when the process is regular enough and,for all t > 0 and all x ∈ X , the measure Pt(x, dy) admits a density, usually denoted bypt(x, y) with respect to some reference measure on X . The theory of Markovian holonomyfields as we develop it restricts itself to fields which are regular enough to ensure that weare in an analogous situation, where all transition kernels have a density.

It is time to describe these kernels. Just as the transition of a usual process alongan interval depends only on the length of this interval and not on its position in the realline, the transition of a Markovian holonomy field along a surface depends on this surfaceessentially only up to diffeomorphism. A surface is characterised by its orientability, whichwe denote by a sign, + or −, by the number of connected components of its boundary,which we denote by p, and by its reduced genus, which we denote by g (see the definitionof this reduced genus in Insert 1). This topological structure is completed by what playsthe role of the length of an interval, namely a total area, denoted by t, of a measure ofarea with which we endow the surface. A theorem of Moser asserts that the only invariantunder diffeomorphisms of such a measure of area is precisely its total area. The quadruple(±, p, g, t) characterises completely the surface up to diffeomorphism and the transitionkernels of a Markovian holonomy field are a family of functions

Z±p,g,t : Gp → R+, (1.3)

analogous to the densities pt(x, y) alluded to earlier.Let us now consider a surface whose boundary has p connected components, with

reduced genus g and total area t. Let us choose q connected components of its boundaryand declare them incoming, whereas the p − q others are declared outgoing. If we fix“initial” conditions, namely q elements x1, . . . , xq ∈ G, along the q incoming boundarycomponents, for each of which we have chosen an orientation, then the field conditionedto satisfy these initial conditions determines a process indexed by the loops traced onour surface and we can in particular consider the distribution of the random variablesassociated with the p − q outgoing connected components, seen as p − q special loops.This distribution is given by

Z±p,g,t(x−11 , . . . , x−1

q , y1, . . . , yp−q) dy1 . . . dyq.

The reference measure dy on the group G is its normalised Haar measure, that is, itsunique Borel probability measure which is invariant under translations. The exponentswhich appear take care of our decision to consider some components as incoming and theothers as outgoing, and make the function Z±p,g,t symmetric in all its arguments.

The usual Markov property, for processes indexed by time, can be expressed by thefact that the transition kernels form a convolution semigroup. From the point of view ofintervals seen as cobordisms between pairs of points, this property governs the behaviourof the process on a time interval which results from gluing together two intervals which


x1

x2

x3

x4

y1

y2

y3

Figure 1.4: Here, p = 7, q = 4 and g = 6 (it is the reduced genus).

have one point in common. In the 2-dimensional context, gluing surfaces along some oftheir boundary components, or gluing two components of the boundary of a single surface,or even gluing with itself a component of the boundary of a surface, are operations whichproduce new surfaces and each of them gives rise to a relation between the transitionkernels of a Markovian holonomy field, analogous to the Chapman-Kolmogorov equation.

If ε and ε′ are two signs, we denote by ε ∧ ε′ the sign which is equal to + if and onlyif both ε and ε′ are +.

Proposition 1.1.6 Let (Z±p,g,t) be the partition functions of a Markovian holonomy field.For all meaningful choice of the parameters, one has the following relations.∫

G

Zεp,g,t(x1, . . . , xp−1, z)Z

ε′

p′,g′,t′(y1, . . . , yp′−1, z−1) dz = (1.4)

Zε∧ε′p+p′−2,g+g′,t+t′(x1, . . . , xp−1, y1, . . . , yp′−1).

This relation corresponds to the gluing of two distinct surfaces along a boundary componentof each.

x1

x2

y1 y1

x1

x2

z z

Figure 1.5: Here, ε = ε′ = +, p = 3, g = 0, p′ = 2 and g′ = 2. The most natural orientation ofthe surfaces being understood, only the boundary component of the two-holed torus labelled by z isnegatively oriented. This accounts for the exponent −1 which it carries in the integral.

∫G

Zεp,g,t(x1, . . . , xp−2, z, z

−1) dz = Zεp−2,g+2,t(x1, . . . , xp−2). (1.5)

This relation corresponds to the gluing of two boundary components of the same surface.Note that our convention on the reduced genus allows us not to distinguish between theorientable case (where the usual genus is increased by 1) and the non- orientable case(where the genus is increased by 2).


x1

z

z

x1

Figure 1.6: Here, ε = +, p = 3, g = 0.

∫G

Zεp,g,t(x1, . . . , xp−1, z

2) dz = Z−p−1,g+1,t(x1, . . . , xp−1). (1.6)

This last relation corresponds to the gluing of one boundary component of a surface withitself, according to a fixed point free involutive orientation-preserving diffeomorphism. Thesurface thus obtained is always non-orientable, homeomorphic to the surface which onewould have obtained by gluing a Mobius band along the boundary component.

Figure 1.7: It is difficult to draw the result of such a gluing. We represent the way in which a connectedcomponent of the boundary of the surface can be glued along the boundary of a Mobius band.

The infinity of transition kernels of a Markovian holonomy field thus satisfy an infinityof relations, one for each operation of surgery. The fact, which we have mentioned earlier,that every compact surface can be obtained by gluing enough disks and three-holed spheres(see Figure 1.8), thus implies that the whole set of transition kernels is in fact determinedby a small number of them.

Figure 1.8: The orientable surface of reduced genus 6 with 5 holes is realised by gluing several three-holed spheres and one disk. Such a decomposition is not unique and the one we have represented doesnot use a minimal number of elementary blocks.

1.2. MULTIPLICATIVE PROCESSES INDEXED BY PATHS 21

Proposition 1.1.7 All transition kernels of a Markovian holonomy field are determinedby the kernels (Z+

1,0,t)t>0 and (Z+3,0,t)t>0, associated respectively to disks and three-holed

spheres of all areas.Moreover, under an assumption of regularity which we will make more explicit later,

the kernels (Z+1,0,t)t>0 suffice to determine all the others.

In the case of the Poisson process indexed by loops, the kernels (Z+1,0,t)t>0, associated

with disks, are the 1-dimensional marginal distributions of the process indexed by R+

which we have described in Proposition 1.1.3. In the general case, we have the followingresult, under certain assumptions of regularity for the Markovian holonomy field.

Proposition 1.1.8 Consider a Markovian holonomy field with values in a group G. Themeasures (Z+

1,0,t(x) dx)t>0 on G form a convolution semi-group on G. They are the 1-dimensional marginal distributions of a G-valued Levy process, which is said to be associ-ated with the field under consideration.

With the unpleasant restriction that it is not proved in all cases that a Markovianholonomy field is completely determined by its transition kernels, the last result establishesa correspondence between Levy processes on a compact Lie group and holonomy fieldswith values in this group. The next result is one of our fundamental results on Markovianholonomy fields. We will state it again, more precisely, at the end of this chapter, afterwe have introduced the necessary definitions.

Theorem 1.1.9 ([L7, Theorem 4.3.1]) Let G be a compact Lie group. For all G-valued Levy process which satisfies certain technical assumptions, there exists a Markovianholonomy field which takes its values in G and to which this Levy process is associated.

The proof of this result of existence is constructive and we will spend most of thischapter explaining how one manages to construct a Markovian holonomy field.

1.2 Multiplicative processes indexed by paths

1.2.1 The measureable space

Let us start by describing more precisely the objects which we consider. We give ourselvesa surface M . It is endowed with its differentiable structure, with an orientation if it isorientable and a Borel measure which allows us to measure areas. We will assume thatthis measure admits, in each coordinate chart, a smooth positive density with respect tothe Lebesgue measure. We will denote it by vol. If M is oriented, one can identify vol witha volume 2-form, that is, a non-vanishing differential 2- form. The class of paths whichwe consider is the class of rectifiable paths, that is, continuous paths with finite length.The fact that a path has finite length is not a metric notion, but indeed a differentiableone: it is invariant under diffeomorphisms. We will denote by P(M) the set of paths withfinite length, taken up to increasing reparametrization. When c is a path, we will denoteby c−1 the same path traced backwards, and c and c the origin and finishing point of c.If c1 and c2 are two paths such that c1 = c2, we denote their concatenation by c1c2.


The group which we are going to consider and in which the field is going to take itsvalues is a compact Lie group, which we denote by G.

INSERT 2 – Compact Lie groups [BtD85, DK00]

A compact Lie group is a compact topological group which is also a differentiable manifold in such a way thatproduct and inversion are smooth mappings. Such a group is isomorphic, by a group isomorphism which isalso a diffeomorphism, to a closed subgroup of the unitary group U(N) for some N . One looses thus onlysome intrinsic character of the description, but no generality, in thinking of a compact Lie group as a compactgroup of matrices with complex coefficients 1. This is what we do from now on, usually without mentioningit, as soon as we feel that it allows us to simplify the presentation. Observe that any finite group endowedwith the discrete topology is a compact Lie group.

Let G be a compact Lie group. It has two properties which are fundamental for us. Firstly, it carries aunique Borel probability measure which is invariant by all translations: the Haar measure. We will not denotethis measure, meaning that we will simply write

∫Gf(g) dg for the integral of a continuous function f : G→ R

with respect to this measure. Secondly, the Lie algebra of G carries a scalar product which is invariant byadjunction. This Lie algebra can be defined as the linear subspace of MN (C) given by

g = A ∈MN (C) : ∀t ∈ R, etA ∈ G.

The real linear space g carries a scalar product (and in general, more than one, even up to multiplication bypositive scalars) invariant under the action of G by adjunction, that is, by conjugation : ad(g)A = gAg−1.The data of a scalar product on g is equivalent to the data of a Riemannian metric on G for which rightand left translations are isometries. The Riemannian volume of such a metric is thus a multiple of the Haarmeasure.

In the case of the unitary group U(N), the Lie algebra u(N) is the space of skew- Hermitian matrices.The scalar product Tr(A∗B) = −Tr(AB) is then an invariant scalar product. The bilinear form Tr(A∗B) +αTr(A∗)Tr(B) is also invariant for all real α, so that it is a scalar product as soon as it is a scalar product,which is the case for α > − 1

N .

We will later need some simple topological considerations on compact Lie groups. If a compact Lie groupG is not connected, then the connected components of the unit element is a normal subgroup of G denotedby G0 such that the quotient G/G0 is finite.

Let us assume that G is connected. The fundamental group of G, as the fundamental group of a topo-

logical group, is Abelian. In fact, given two continuous loops γ1, γ2 : [0, 1] → G based at the unit element,

the concatenation γ1γ2 of these two loops is homotopic to their pointwise product t 7→ γ1(t)γ2(t), as well

as to their pointwise product in the other order: t 7→ γ2(t)γ1(t). The pointwise multiplication endows free

homotopy classes, without any conditions on the endpoints, with the structure of an Abelian group canonically

isomorphic to π1(G, 1), where 1 denotes the unit element. This group is finite if and only if the centre of G is

finite, in which case G is said to be semi-simple. In this case, G admits a universal covering by another compact

Lie group. For example, the fundamental group of SO(3) is isomorphic to Z/2Z and SO(3) admits a universal

covering, of degree 2, by SU(2). In the case where the centre of G is not finite, for example if G = U(N),

then a universal covering of G is isomorphic to the product of a compact Lie group and Rn for a certain n ≥ 1.

Let us describe the canonical space of a multiplicative process indexed by a subset ofP(M) and with values in G.

1Since U(N) can be seen as a subgroup of SO(2N), one can just as well think of matrices with real coefficients.


Definition 1.2.1 Let P be a subset of P(M). Let G be a compact Lie group. A functionh : P → G is said to be multiplicative if

1. ∀c ∈ P, c−1 ∈ P ⇒ h(c−1) = h(c)−1.2. ∀c1, c2 ∈ P, c1c2 ∈ P ⇒ h(c1c2) = h(c2)h(c1).

We denote by M(P,G) the set of multiplicative functions from P into G.

We are going to endow M(P,G) with the σ-field C generated by the evaluation map-pings h 7→ h(c), for c running through all of P . A random holonomy field on M is a finiteor probability measure on (M(P(M), G), C).

1.2.2 Graphs

In order to construct a measure on (M(P(M), G), C), we would like to follow the usualprocedure for constructing a stochastic process, by first describing the finite-dimensionalmarginals and then applying a Kolmogorov theorem. This is unfortunately not possibledirectly here, because we are not able to write down in a simple way all the finite-dimensional distributions of the measures which we want to construct. Before we explainwhere this problem comes from, let us first describe those finite-dimensional marginalsfor which it is possible.

We will say that a loop l : [0, 1]→M is simple if it is injective on [0, 1).

Definition 1.2.2 A graph on M is a triple G = (V,E,F) such that1. E is a set of edges, that is, injective paths or simple loops, which do not meet the

boundary of M or are contained in it; we assume that E contains the edge e−1 as soonas it contains the edge e, and that two distinct edges which are not each other’s inversemeet, if at all, only at some of their endpoints,

2. V is the set of endpoints of the edges of E,3. F is the set of connected components of the complement in M of the union of the

ranges of the edges of E,4. each element of F is homeomorphic to an open disk of R2.

The elements of V are called vertices of the graph, and the elements of F its faces.

Let G be a graph on M . Let us denote by P(G) the set of paths that can be con-structed by concatenating edges of G. We are going to describe a probability measureon M(P(G), G). Since P(G) is infinite, this is not properly speaking a finite-dimensionalmarginal of the holonomy field, but it is easy to check that the restriction mappingM(P(G), G) → M(E, G) is a bijection. In other words, a multiplicative mapping onpaths traced in a graph is completely determined by its values on the edges. If we choosemoreover an orientation of G, that is a subset E+ of E such that E is the disjoint unionof E+ and (E+)−1, then it suffices to know a multiplicative function on E+: there is alsoa bijective restriction mapping M(E, G)→M(E+, G). Finally, any function from E+ toG is multiplicative, because E+ contains the inverse of none of its elements, nor any oftheir concatenations. Thus, M(E+, G) = GE+

.

The measure dg⊗E+

, which is the Haar measure on the compact Lie group GE+, de-

termines through the bijections which we have just described measures on M(E, G) and


Figure 1.9: A graph on a three-holed sphere. The assumption that the faces are homeomorphic to disksimplies that the boundary of the surface is covered by edges of the graph, in the present case two injectiveedges and two simple loops.

M(P(G), G), which we call the uniform measures and denote simply by dh. The measurewhich we are going to define on M(P(G), G) has a density with respect to this uniformmeasure, which depends on the choice of a Levy process on G.

1.2.3 Levy processes and marginals associated with graphs

Let us start by briefly describing Levy processes on G.

INSERT 3 – Levy processes on a compact Lie group[Lia04]

A process X = (Xt)t≥0 with values in G is said to be a Levy process if it is cadlag and if its incrementsare independent and stationary. Since the multiplication in G is not commutative, one needs to choose aside in order to defined increments, and we decide that the increment of X between s and t is X−1s Xt

(one speaks of a left Levy process). Let X be a Levy process. Then the formula Ptf(x) = E[f(xX−10 Xt)]defines a left-invariant Feller semigroup, in the sense that for all x, y ∈ G and all Borel subset Γ of G, onehas Pt(yx, yΓ) = Pt(x,Γ). Conversely, any homogeneous Feller process on G whose transition kernel isleft-invariant is a left Levy process. Finally, the data of a Levy process is equivalent to the data of weaklycontinuous convolution semigroup of probability measures on G, namely the semigroup of the 1-dimensionalmarginal distributions of X.

The generator of a Levy process is the sum of a diffusive term, a drift term and a jump term, the lattercharacterised by a measure denoted by Π, which does not charge the unit element and satisfies the condition∫GdG(1, x)2 Π(dx) < +∞, where dG is a bi-invariant Riemannian distance on G. We are going to consider

processes issued from the unit element and invariant by conjugation, that is to say such that for all g ∈ G,the processes X and gXg−1 have the same distribution. For such processes, the three parts of the generatorare invariant by conjugation. This constraint is very strong on the diffusion and drift part : if the group G issimple, for example if G = SU(N), the the drift must be zero and the diffusive part of the generator is fullyprescribed by the condition of invariance, up to multiplication by a non-negative constant.

We need to consider Levy processes whose distribution at each positive time has a continuous density withrespect to the Haar measure. The following result shows that this happens under an apparently much weakerassumption. Recall that G0 is the connected component of the unit element in G and G/G0 is finite.

Proposition 1.2.3 ([L7, Proposition 4.2.3]) Let X be a Levy process invariant by conjugation on G, issuedfrom the unit element. If there exists p > 1 such that for all t > 0 the distribution of Xt admits a densitywhich is in the Lp space of the Haar measure on G, then this density, which we will denote by (t, x) 7→ Qt(x),


is continuous on (0,+∞)×G and positive on (0,+∞)×G0. If moreover the support of the image measureof Π by the projection G→ G/G0 generates G/G0, then Q is positive on (0,+∞)×G.

Definition 1.2.4 We will say that a Levy process X on G is admissible if it is issuedfrom the unit element, invariant in distribution by conjugation and if for all t > 0 thedistribution of Xt admits a density Qt with respect to the Haar measure, such that (t, x) 7→Qt(x) is continuous and positive on (0,+∞)×G.

The group G admits a Riemannian metric such that the translations are isometries(see insert 2). Such a metric is not unique, but we choose one and we denote by dG theassociated Riemannian distance. Most of the time, we will write distances between theunit element and another element of G. This distance can be replaced without damagein all formulas by a Euclidean distance in MN(C) between, the identity matrix and theelement under consideration.

Let us give examples of admissible processes.

• If G is a finite group, then the admissible processes are exactly the random walkswhose jump measure is invariant by conjugation and has a generating support.

• If G is connected, the the Brownian motion on G is an admissible Levy process.

• If X is an admissible Levy process, one can add or subtract to its Levy measure anyfinite measure supported by G0 without changing the fact that it is admissible.

• There exist pure jump admissible processes, like for example the process on SU(2)whose jump measure is Π(dx) = dG(1, x)α dx for α ∈ (−5,−3).

Let X be an admissible Levy process on G, of which we denote by Q the density.Let M be a surface endowed with a graph G. We are now going to define a measure onM(P(G), G) associated to X.

Observe that each face of G is bounded by a loop in G. This apparently trivial remarkis not completely straightforward : firstly, the loop is not unique, since it is defined upto orientation and choice of an origin, and secondly it is not necessarily a simple loop,since it can take twice the same edge, possibly twice in the same direction if the surfaceis not orientable (see Figure 1.10). In order to simplify the exposition, we will ignore thenon-orientable case in most of the forthcoming statements. In an orientable and orientedsurface, the boundary of a face F is thus a loop ∂F defined up to the choice of its origin.Thus, if h : P(G) → G is a multiplicative function, the value of h(∂F ) is defined up toconjugation. This is enough for the next definition to make sense.

Definition 1.2.5 Let M be a surface endowed with a measure of area denoted by vol. LetG be a graph on M . One defines the probability measure DFX,GM,vol on (M(P(G), G), C) by

DFX,GM,vol(dh) =1

ZX,GM,vol

∏F∈F

Qvol(F )(h(∂F )) dh.


It is possible to incorporate boundary conditions in the definition of DFX,GM,vol but weprefer not do so in order keep the exposition as simple as possible. Let us simply indicatethat the density is not affected by the boundary conditions, only the uniform measuredh is conditioned to satisfy the desired boundary conditions. The normalisation constantZX,GM,vol is then modified and becomes a function of the boundary conditions.

Let us also mention that, in the non-orientable case, it is also necessary to assumethat the process X is invariant by inversion, that is, that it has the same distribution asX−1.

Figure 1.10: This graph on a Mobius band has two vertices, three edges and a single face, whoseboundary goes twice through the equator (the bolder edge), in the same direction.

We are now going to justify, to the extent possible, the definition 1.2.5, by finding asense in which it is natural. For this, and in the first place, we discuss gauge symmetry.

1.2.4 Gauge symmetry

Definition 1.2.6 The gauge group is the group J = F(M,G) of all functions from M toG. If P is a subset of P(M), then J acts on M(P,G) in the following way : the elementj of J transforms the element h of M(P,G) in the element j · h defined by

∀c ∈ P , (j · h)(c) = j(c)−1h(c)j(c).

For geometric of physical reasons which we will discuss later, one should consider onM(P(M), G) only objects, measures or functions, which are invariant under the action ofthe gauge group. This restricts a great deal the space of observables, that is the space ofpermitted functions. For example, if c is a path whose extremities are distinct, then theonly functions f : G → R such that h 7→ f(h(c)) is invariant under the action of J areconstant functions. Indeed, the value of h(c) can be transformed into any element of G byan appropriate gauge transformation. On the other hand, if c is a loop, the any functionf : G → R which is invariant by conjugation, that is such that f(yxy−1) = f(x) for allx, y, gives rise to a function h 7→ f(h(c)) which is invariant under gauge transformations.

A closer examination indicates that gauge transformations conjugate the values of hon loops, but in fact conjugates by the same element the values of h on loops based atthe same point. Let us define the diagonal action of G on Gn for all integer n ≥ 1 byy · (x1, . . . , xn) = (yx1y

−1, . . . , yxny−1). Then for all function f : Gn → R invariant by


diagonal conjugation and all family l1, . . . , ln of loops based at the same point, the functionh 7→ f(h(l1), . . . , h(ln)) is invariant under gauge transformations and it is essentially themost general one.

Before we state a result, let us give an example of diagonal conjugacy classes. In thegroup SO(3), each rotation which is neither the identity nor an axial symmetry (let uscall such rotations proper rotations) has an axis, which is an oriented line, and an angle,which belongs to (0, π). This angle characterises the conjugacy class of the rotation. Letus now consider two collections (r1, . . . , rn) and (r′1, . . . , r

′n) of proper rotations. Then ri

is conjugated to r′i for all i = 1, . . . , n if and only if ri and r′i have the same angle forall i = 1, . . . , n. However, there exists a single rotation r such that r′i = rrir

−1 for alli = 1, . . . , n if and only if, in addition to the previous condition, there exists a rotation(which is none other than r) which, for all i = 1, . . . , n, sends the oriented axis of ri tothe oriented axis of r′i. Thus, the diagonal conjugacy class of a family of rotations is givenby the angles of these rotations and the relative position of their axes.

Proposition 1.2.7 Let P be a subset of P(M) stable by inversion and by concatenation.Assume that each extremity of a path of P is joined to any other by a path of P . Let mbe an extremity of a path of P . Let Lm be the set of loops based at m which belong to P .Then the following two assertions hold.1. The composed mapping M(P,G) −→ M(Lm, G) −→ M(Lm, G)/G, where the firstmapping is the restriction and the second is the quotient by the action of G by diagonalconjugation on all factors, can be factored into an injection

M(P,G)/J −→M(Lm, G)/G.

2. Let c1, . . . , cn be elements of P . Let f : Gn → R be a continuous function such that thefunction h 7→ f(h(c1), . . . , h(cn)) is invariant under gauge transformations. Then there

exists n loops l1, . . . , ln based at m and a continuous function f : Gn → R invariant underthe diagonal action of G and such that for all h ∈M(P,G) one has f(h(c1), . . . , h(cn)) =

f(h(l1), . . . , h(ln)).

When one restricts to observables invariant by gauge transformations, one loses nothingin restricting oneself to loops, nor even to loops based at a single arbitrary point.

The proposition above makes hopefully the next definition understandable, of a σ-fieldon M(P,G) which is smaller than the cylinder σ-field which we have considered so far.

Definition 1.2.8 Let P be a subset of P(M) stable by inversion and by concatenation.Assume that each extremity of a path of P is joined to any other by a path of P . Onecalls invariant σ-field, and one denotes by I, the smallest σ-field which makes all functionsh 7→ f(h(l1), . . . , h(ln)) measurable, where l1, . . . , ln are loops in P based at the same pointand f : Gn → R is invariant under the diagonal action of G by conjugation.

Let us come back to the probability measure that we have defined on M(P(G), G).The next result should come as no surprise.

Lemma 1.2.9 The measure DFX,GM,vol is invariant under the action of the gauge group.


Let us conclude this paragraph by an observation on the nature of the boundary con-ditions which can be imposed to a holonomy field. When one puts boundary conditions tothe measure DFX,GM,vol, one prescribes the value of multiplicative functions along connectedcomponents of the boundary of M . These connected components have on one hand noprivileged origin and, on the other hand, would they have one, that these would neces-sarily be distinct. There is thus no meaning, up to gauge transformation, to prescribemore than the conjugacy class of the value of the functions along each component of theboundary. The boundary conditions thus consist of the data of one conjugacy class of Gfor each oriented connected component of the boundary of M (see Definition 1.3.2).

1.2.5 A random representation of the group of loops

The gauge invariance of the measure DFX,GM,vol allows us to see it as a good object from thegeometrical and physical point of view, but it does not suffice to make it look natural.In order to understand it better, let us study the structure of the set of loops based at apoint in a graph.

INSERT 4 – Group of loops in a graph

Let G be a graph. Let m be a vertex of G and let Lm(G) be the set of loops in G based at m. The con-catenation endows Lm(G) with a structure of monoid. Let us declare that two loops are equivalent if onecan transform one into the other by a sequence of insertions or erasures of paths of the form ee−1 where e isan edge of G. This relation is compatible with concatenation and the quotient of Lm(G) by this equivalencerelation is a group. Let us called reduced loop a loop which contains no sub-path of the form ee−1. Thenone proves that each equivalence class contains a unique reduced loop, which is also the unique element ofshortest length in this class. One can thus choose to represent each class by its unique reduced representative,and consider as group operation the operation of concatenation- reduction. One obtains in this way a group,denoted by RLm(G), which is canonically isomorphic to the fundamental group π1(M \ Y,m) where Y is afinite subset of M which contains exactly one point in each face of G.

The group RLm(G) is free and the simplest way to exhibit a base of this group is to choose a spanningtree of G and to associate to each edge not covered by this tree a loop of RLm(G). For this, to each edge ewhich is not in the spanning tree, one associates the loop which starts from m, goes to the origin of e by theunique injective path which does so in the tree, then traverses e and finally goes back from the endpoint of eto m injectively along the tree.

Let v = #V, e = #E/2 and f = #F be the number of vertices, edges and faces of G. Let also g denotethe reduced genus of M (that is twice the usual genus of the surface obtained from M by gluing a disk alongeach component of the boundary of M) and p the number of boundary components of M . Then there aree− v + 1 edges not covered by a spanning tree of G. The Euler characteristic of M is equal to 2− g − p onone hand and to v − e + f on the other hand. Thus, the number of edges not covered by a spanning tree ofG is g + p + f − 1.

The group RLm(G) is thus a free group of rank g + p + f − 1. On a disk for example, this number equalsf, the number of faces of G. It is possible to give one, indeed many bases of this group which are explicitlyin one-to-one correspondence with the set of faces of G. Let us call lasso a loop which is of the form sbs−1,where s is a path issued from m and b is a simple loop based at the endpoint of s. The loop b is called themeander (or buckle) of the lasso. We say that a lasso is facial if its meander is the boundary of a face of G.For a graph on a disk, it is possible to find bases of RLm(G) formed by facial lassos, one for each face of G,in such a way that the product of these lassos in a certain order is a lasso whose meander is the boundary ofthe disk. In the general case, the result is the following. Although RLm(G) is free, it is more natural to give apresentation of it which involves one relation, just as the space of finite real sequences (a0, . . . , an) with sumequal to 0, although n-dimensional, is more naturally described as the kernel of a linear form in Rn+1 than asthe range of a linear mapping Rn → Rn+1.


Proposition 1.2.10 Let M be a connected surface with reduced genus g, whose boundary has p connectedcomponents. Let G be a graph on M with f faces. Let m be a vertex of G. There exists g lassos a1, . . . , ag,a word w in g letters and their inverses, p lassos c1, . . . , cp, the meanders of which are the p connectedcomponents of the boundary of M , and f facial lassos l1, . . . , lf bounding the f faces of G, such that the groupRLm(G) admits the presentation

〈ai, cj , lk | w(a1, . . . , ag)c1 . . . cpl1 . . . lf = 1〉,

and, for all continuous function f : Gg+p+f → R, one has∫M(P (G),G)

f(h(a1), . . . , h(ag), h(c1), . . . , h(cp), h(l1), . . . , h(lf)) dh

=

∫Gg+p+f

f(x1, . . . , xg, y1, . . . , yp, z1, . . . , zf−1, zf)dxidyjdzk,

where we have set zf = (w(x1, . . . , xg)c1 . . . cpz1 . . . zf−1)−1.

The lassos a1, . . . , ag generate in particular, with the relation w(a1, . . . , ag) = 1, the fundamental groupof surface obtained from M by gluing a disk along each boundary component.

With boundary conditions specified, the variables h(ci) would be uniform on the conjugacy classes which areprescribed to them. In the present case, we see that, under the uniform measure, the variables h(l1), . . . , h(lf)are just as independent as the topology of M allows them to be.

Consider a multiplicative function h : P(G)→ G. We know by Proposition 1.2.7 that,up to gauge transformation, h is characterised by its restriction to Lm(G), where m is avertex of G chosen arbitrarily. It follows immediately from the multiplicativity propertyof h that two loops which are equivalent in the sense defined in the last insert have thesame image by h. Thus, h determines a multiplicative mapping from RLm(G) to G, whichis nothing but a group homomorphism from RLm(G) into G. We can write, somewhatabstractly,

M(P(G), G)/J ' Hom(RLm(G), G)/Int(G),

where Int(G) is the group of inner automorphisms of G, those which are given by theconjugation by a fixed element, which acts on the left by composition on Hom(RLm(G), G).

Thus, we see that the measure DFX,GM,vol determines a random homomorphism fromRLm(G) into G whose distribution is invariant by composition by an inner automorphism.The distribution of such a random homomorphism is characterised by the distribution ofthe image of a generating set of RLm(G), which can be chosen to be a basis in order toavoid any kind of algebraic constraint on the image of this generating set, apart from thefact that it must be invariant by diagonal conjugation.

With the description that we have given of RLm(G), and given the Levy process X

on G, the random homomorphism determined by the measure DFX,GM,vol is as natural asit can be. On a disk for instance, it sends the lassos l1, . . . , lf of a basis described byProposition 1.2.10 on independent random variables with respective distributions equalto those of Xvol(F1), . . . , Xvol(Ff), where of course the faces have been numbered in sucha way that the lasso li bounds the face Fi. This description of the distribution of therandom homomorphism depends on the basis of RLm(G) which one chooses and it is not


obvious that two different bases give rise, in distribution, to the same homomorphism.This can be checked directly as a consequence of a theorem of E. Artin on the action ofthe braid group on the free group [Art47].

On surfaces of higher genus, or with boundary conditions specified, the distributions ofthe images of the facial lassos are modified in the most natural possible way. For example,on a disk with boundary condition given by an element x1 of G, one conditions the Levyprocess X to reach the conjugacy class of X at a time equal to the total area of the disk,and the images of l1, . . . , lf have the distribution of the increments of this process. Anothersimple example is that of a sphere, where the process X is conditioned to come back to theunit element at a time equal to the total area of the sphere. One sees incidentally that onefinds something very similar to what happens on a disk with boundary condition given bythe unit element. Finally, one can for example describe what happens on a closed surfacewith reduced genus equal to 4: if the graph G is fine enough, one can choose the basisso that the word w in 4 letters and their inverses is aba−1b−1cdc−1d−1. Then the faciallassos are sent to variables which have the distribution of the increments of the process Xconditioned to have, at the time equal to the total area of the surface, the distribution ofUV U−1V −1WZW−1Z−1, where U, V, Z,W are uniform and independent on G. For somegraphs, the word w can be different, for instance abcda−1b−1c−1d−1, but one checks thatthe law of w(U1, U2, U3, U4) where U1, U2, U3, U4 are uniform and independent on G doesnot depend on the graph G, but only on the topology of M .

1.2.6 Wilson loops [L3]

We are going to make a pause in the progression towards the definition of Markovianholonomy fields and to present a result which is concerned with the discrete theory thatwe have just described. This section can be skipped without damage for the understandingof the main construction.

Let G be a graph on a surface M , although really the surface plays no role here.On the space M(P(G), G), considerations of gauge symmetry have led us to introducefunctions of the form h 7→ f(h(l1), . . . , h(ln)) where l1, . . . , ln are loops based at the samepoint and f is invariant by diagonal conjugation. Propositions 1.2.7 and 1.2.10 allow usto check that these functions separate the points of the quotient space M(P(G), G)/J .Thus, from the physical point of view, they generate a good algebra of observables.

However, in the study of discrete gauge theories, physicists often consider a muchsmaller algebra of observables, generated by simpler functions called Wilson loops. Theyare the functions of the form h 7→ f(h(l)) where l is a loop and f is invariant by conjuga-tion. The natural question which arises is whether these observables suffice to generateall continuous functions on the quotient space M(P(G), G)/J , that is, according to thetheorem of Stone-Weierstrass, whether they separate the points.

Let h be a multiplicative function. The question is the following : knowing the con-jugacy class of h(l) for all loop l, does one know h up to gauge transformation, that is,does one know the class of diagonal conjugation of (h(l1), . . . , h(ln)) for all n- tuple ofloops based at the same point ? Given l1, . . . , ln, we have access to the conjugacy classesof h(l1), . . . , h(ln), but also to those of h(l1l2) = h(l2)h(l1), h(l1l2l3) = h(l3)h(l2)h(l1), oractually of the image by h of any other loop that can be formed by concatenating copies


of l1, . . . , ln and their inverses. We have thus access to the conjugacy class of any productof the elements h(l1), . . . , h(ln) and their inverses of G.

The question is now purely algebraic, in the group in which h takes its values. If thisgroup is SO(3), one can put the question in the following way : given n rotations r1, . . . , rnof R3, does the data of the angle of any rotation that can be formed by multiplying elementstaken, with possible repetitions, in the set r1, . . . , rn, r

−11 , . . . , r−1

n suffice to determinethe relative position of the axes of r1, . . . , rn ?

In this case, the answer is positive.

Theorem 1.2.11 ([L3, Proposition 3.4]) Let G be a finite product of groups taken inthe following list : unitary, special unitary, orthogonal, special orthogonal, symplectic.Let r ≥ 1 be an integer. Let g = (g1, . . . , gr) and g′ = (g′1, . . . , g

′r) be two elements of Gr.

Assume that for all word w in r letters and their inverses, the elements w(g) and w(g′)of G are conjugated. Then there exists k ∈ G such that for all i between 1 and r one hasg′i = kgik

−1.

This theorem has been proved by B. Durhuus for unitary and special unitary groups[Dur80] and by A. Sengupta for unitary and odd orthogonal groups [?]. The proof ofthe theorem above relies on the use of spin networks, which are a natural set of invariantfunctions2 on M(P(G), G) of a different nature and for which it is straightforward tocheck that they generate a dense subalgebra of functions [Bae96]. One then proves, usingthe fundamental theorems of invariant theory, that these spin networks can be expressedin terms of Wilson loops.

1.2.7 Invariance by subdivision

Let us come back to the mainstream of the construction of Markovian holonomy fields.The normalisation constant ZX,G

M,vol which appears in Definition 1.2.5 and which, in presenceof boundary conditions, is a function of these boundary conditions, will be, as the notationsuggests, one of the transition kernels of the holonomy field associated to X. It seemshowever to depend on the graph G. A first important results states that it is not the case.

Proposition 1.2.12 ([L7, Prop. 3.2.5, Prop. 4.1.10]) The number ZX,GM,vol which, in

presence of boundary conditions is a function ZX,GM,vol(x1, . . . , xp) of these conditions, does

not depend on the graph G and is denoted by ZXM,vol(x1, . . . , xp). If M is orientable with

reduced genus g, the one has the following expression :

ZXM,vol(x1, . . . , xp) =

∫G

Qvol(M)([a1, a2] . . . [ag−1, ag]c1x1c−11 . . . cpxpc

−1p ) daidck,

where we have used the notation [a, b] = aba−1b−1.If M is non-orientable with genus g, then

ZXM,vol(x1, . . . , xp) =

∫G

Qvol(M)(a21 . . . a

2gc1x1c

−11 . . . cpxpc

−1p ) daidck.

2They are actually essentially the invariant polynomial functions on M(P(G), G).


The expressions which appear here are reminiscent of the classical presentations of thefundamental groups of surfaces. This is not surprising since these presentations can becomputed using polygonal models for these surfaces, the edges of which are identified bypairs, and that the same polygonal models produce graphs with a single face on thesesurfaces, in which the partition function can be computed to give almost directly theexpressions above.

Now, we would like to bind all the measures DFX,GM,vol into a single measure onM(P(M), G).We have said at the beginning of Section 1.2.2 that this does not work directly. On onehand, the measures DFX,GM,vol do not suffice to describe all the finite-dimensional marginalsof a measure onM(P(M), G): in general, a finite number of paths, or even one singe path,are not paths which can be realised by concatenating the edges of a graph. For example,even a very regular path can have a range whose complement in M has infinitely manyconnected components. Another aspect of the same problem is the fact that the naturalpartial order on the set of graph does not have the required property in order to apply atheorem of the kind of Kolmogorov’s theorem. The partial order is the following.

Definition 1.2.13 A graph G2 is said to be finer than a graph G1 if E1 ⊂ P(G2) or,equivalently, if P(G1) ⊂ P(G2).

The set of all graphs endowed with this partial order is not directed, which means thatgiven two graphs, there does not always exist a third graph which is finer than the firsttwo.

One solves these problems by temporarily restricting the class of paths which oneconsiders on M . The point is to consider paths which, like piecewise affine or real analyticpaths on R2 for instance, have a rigidity property which prevents them from intersectingor self-intersecting in a pathological way. The only differentiable structure on M does notallow one to define such a class. It is thus necessary to enrich the structure of M , keepingin mind that the final object that we construct must be independent of this enrichment.We choose to endow M with a Riemannian metric, which we choose in such a way thatthe Riemannian volume coincides with the measure of area. We denote by A(M) the setof piecewise geodesic paths on M , or by Aγ(M) if we want to refer explicitly to the metric.

Given two graphs G1 and G2 such that G2 is finer than G1, one has a restrictionmapping M(P(G2), G) → M(P(G1), G) induced by the inclusion P(G1) ⊂ P(G2). Thefundamental result of the discrete theory is the following property of invariance by subdi-vision. It is proved in [?], mainly at the proposition 4.3.4.

Proposition 1.2.14 1. Let G1 and G2 be two graphs such that G2 is finer than G1. Thenthe restriction mapping M(P(G2), G)→M(P(G1), G) sends the measure DFX,G2

M,vol on the

measure DFX,G1

M,vol.2. Let γ be a Riemannian metric on M . The family of probability spaces(

M(P(G), G), I,DFX,GM,vol

): G piecewise geodesic graph

is a consistent family. One can take its inverse limit and obtain a probability space(M(Aγ(M), G), I,DFXM,vol), so that the marginal distribution of the canonical process on

this space associated to a graph with piecewise geodesic edges G is the measure DFX,GM,vol.


It is in the proof of this property that the property of semigroup of the 1-dimensionaldistributions of X plays its role. Let us illustrate this on an example. Consider the twographs represented on Figure 1.11. The graph G1 is obtained from G2 by removing theedge e. In G2, this edge is located on the boundary of two faces F1 and F2, one boundedby the loop e1e2e and the other by the loop e−1e3e4. These faces are reunited in G1 toform a single face F , bounded by the loop e1e2e3e4.

e1e2

ee3

e4

e1e2

e3

e4

G2 G1

F1

F2

F

Figure 1.11: Removing an edge to a graph is the most important elementary operation which allowsone to go from one graph to a less fine one.

Let f :M(P(G1), G)→ R be a function. It induces a function f :M(P(G2), G)→ R,

such that the value f(h) of f on a multiplicative function h does not depend on the

value of h(e). The invariance by subdivision means that∫M(P(G2),G)

f(h) DFX,G2

M,vol(dh) =∫M(P(G1),G)

f(h) DFX,G1

M,vol(dh). If we choose an orientation of G2 (as we have partially done

on the picture), we can see the integral on the right as an integral on GE+1 and that on the

left as an integral on GE+2 . There is thus one more variable in the integral on the left, which

corresponds to the edge e. Only the density of DFX,G2

M,vol depends on this extra variable,through the two terms associated to the two faces bounded by e: there is in the integral aterm of the form

∫. . . Qvol(F1)(h(e)h(e2)h(e1))Qvol(F2)(h(e4)h(e3)h(e)−1) . . . d(h(e1)) which,

by renaming the variables h(e1), h(e2), h(e3), h(e4), h(e) as x1, x2, x3, x4, x and using theinvariance by conjugation of Q, can be rewritten perhaps more suggestively as

∫. . . Qvol(F1)(x2x1x)Qvol(F2)(x

−1x4x3) . . . dx.

The property of convolution semigroup of Q allows us to integrate with respect to x, thusproducing the term Qvol(F1)+vol(F2)(x2x1x4x3) = Qvol(F )(x4x3x2x1), which is exactly the

term corresponding to F in the density of DFX,G2

M,vol.

Once the invariance by subdivision is established, one can apply a theorem of the kindof Kolmogorov’s theorem, in a version which is not very usual because of the size of theindex space, but actually easier to prove than the more common versions, thanks to thecompactness of G.


1.2.8 Extension to rectifiable loops

At the cost of an artificial supplement of structure on M , we have now at our disposala process with values in G indexed by piecewise geodesic loops. In order to go frompiecewise geodesic loops to rectifiable ones, we are going to proceed by approximation,and this requires that we endow P(M) with an appropriate topology.

INSERT 5 – Topology on the space of paths

The data of a Riemannian metric on M makes it a metric space (M,d) and allows one to measure the lengthof rectifiable paths. We will denote by `(c) the length of a path c. We define a distance on P(M) by setting,for all c, c′ parametrized at constant speed,

d`(c, c′) = maxd(c(t), c′(t)) : t ∈ [0, 1]+ |`(c)− `(c′)|.

The topology of the metric space (P(M), d`) does not depend on the Riemannian metric chosen on M . Thisdistance is easy to manipulate, but it has the unpleasant property that the space (P(M), d`) is never complete.Indeed, it is easy to construct a sequence of paths of length 2 which converges uniformly while zigzaggingtowards a geodesic segment of length 1.

In fact, there exists another distance on P(M), which is complete and metrized the same topology as d`.It is the distance in 1-variation which, on R2 for example reads

d1(c, c′) = maxd(c(t), c′(t)) : t ∈ [0, 1]+

∫ 1

0

|c(t)− c′(t)| dt.

This distance is however much less easy to handle than d` and it matters to us mainly insofar as it gives us acomplete metrisation of the topology that we use.

Most of the time, because of the invariance under gauge transformations, we will consider convergenceswith fixed endpoints : we will say that a sequence (cn)n≥0 of paths converges towards a path c with fixedendpoints if d`(cn, c) tends to 0 and if, for all n ≥ 0, the path cn has the same endpoints as c.

The set A(M) of piecewise geodesic paths is of course dense in P(M) for the convergence with fixedendpoints.

In order to define a measure on M(P(M), G), we need two things : a property ofregularity of the measure DFXM,vol on M(A(M), G) and a theorem of extension. Let usstart with the first thing. Recall that dG is the Riemannian distance on G for a bi-invariantmetric.

Proposition 1.2.15 Consider the measure DFXM,vol onM(A(M), G). There exists K > 0

such that for all simple loop l ∈ A(M) bounding a disk D and such that `(l) < K−1, onehas ∫

M(A(M),G)

dG(1, h(l)) DFXM,vol(dh) ≤ K√

vol(D).

This property, which expresses the fact that the holonomy along the boundary of asmall disk is close to the unit element, can only be inherited from a similar property ofthe L evy process X. The following property is in fact true for a Levy process on anarbitrary Lie group.


Proposition 1.2.16 ([Lia04, Lemma 3.5]) Let X = (Xt)t≥0 be a Levy process issuedfrom the unit element on G. There exists a constant K such that

∀t ≥ 0, E[dG(1, Xt)] ≤ K√t.

In order to use this regularity property of DFXM,vol, we are going to consider the group Γ

of all random variables with values inG defined on the probability space (M(A(M), G),DFXM,vol).This group is endowed with a distance δ defined by the formula

∀S, T ∈ Γ, δ(S, T ) = E[dG(S, T )].

The pair (Γ, δ) is a complete metric topological group on which the properties of invarianceof dG imply that the translations and the inversion are isometries.

We are going to consider the canonical process on (M(A(M), G),DFXM,vol) as a mul-tiplicative function from A(M) into Γ: to each path c is associated the random variableHc defined by Hc(h) = h(c). The regularity property stated in Proposition 1.2.15 can berewritten as

δ(1, Hl) ≤ K√

vol(D).

We apply now the following result.

Theorem 1.2.17 ([L7, Theorem 3.3.1]) Let (M,γ) be a compact Riemannian surface.Let vol denote the Riemannian volume of γ. Let (Γ, δ) be a complete metric group onwhich translations and inversion are isometries. Let H ∈M(Aγ(M),Γ) be a multiplicativefunction. Assume that there exists K > 0 such that for all simple loop l ∈ Aγ(M) bounding

a disk D and such that `(l) ≤ K−1, the inequality δ(1, H(l)) ≤ K√

vol(D) holds.Then H admits a unique extension to an element of M(P(M), G), also denoted by H,

such that if a sequence (cn)n≥0 of paths converges with fixed endpoints to a path c, thenH(cn) −→

n→∞H(c).

The multiplicative function H : P(M) → Γ determines, following backwards whatwe have done a few paragraphs above, a probability measure on M(P(M), G), which wedenote by HFXM,vol. The following theorem, the proof of which is unfortunately not verypleasant, guarantees that we have reached our goal.

Theorem 1.2.18 ([L7, Proposition 3.4.1]) The measure HFXM,vol does not depend onthe choice of the Riemannian metric on M . Moreover for all graph G on M , whose edgesneed not be piecewise geodesic for any Riemannian metric, the marginal distribution ofHFXM,vol associated to P(G) is the measure DFXM,vol.

INSERT 6 – The Banchoff-Pohl inequality [BP72, Vog81]

Let us give an application of Theorem 1.2.17 in a simpler and more familiar setting. Let us work in the planeR2. Recall that we have defined, for all continuous loop l : [0, 1]→ R2, the function nl outside the image of l(see (1.1)). Consider the usual Euclidian metric on R2 and denote by A(R2) the set of piecewise affine paths.If the loop l is piecewise affine, then the complement of its range has finitely many connected components


and the function nl, locally constant, takes only a finite number of values. It is in particular bounded and,since it has compact support, square-integrable.

To each piecewise affine path c, let us associate the loop obtained by concatenating to c the two segmentswhich join the origin of R2 to each of its endpoints and let us use the same notation nc for the functionassociated to this loop. With this notation and considering the discussion above, one checks immediately thatthe mapping c 7→ nc is a multiplicative (or rather, additive) function

n : A(R2)→ L2(R2).

If l is a simple loop which bounds a disk D, we have the equality ‖nl‖L2 =√

area(D).In order to really be in the setting of Theorem 1.2.17, we should restrict ourselves to a compact disk of

the plane, but we take this disk large enough to simply ignore that it is not the whole plane. We can thusclaim that the function c 7→ nc extends to a continuous function from P(R2) to L2(R2).

In particular, we deduce that for all rectifiable loop l, the function nl is square- integrable. Moreover, itdepends continuously (in the L2 sense) on the loop l (in the sense of the convergence for the distance d` orthe convergence in 1- variation).

The fact that nl is square-integrable for all rectifiable loop is known, if not well- known, and it is quantifiedby the Banchoff-Pohl inequality, which generalises the usual isoperimetric inequality by stating the following :

‖nl‖2L2 ≤ 1

4π`(l)2. (1.7)

1.3 Markovian holonomy fields

1.3.1 Definition

The construction which leads to the definition of the measure HFXM,vol considers onlyone surface, with possibly a set of boundary conditions which are determined from thebeginning. In order to go from this point of view to the definition of Markovian holonomyfields, we need only one more step, in which we are going to consider all possible surfacesat once, and relate them thanks to the operations of cutting and gluing. In order to givea precise definition, we need to give a more formal framework to boundary conditions andgluing operations.

When one glues together two surfaces along two components of their boundary, alongwhich compatible boundary conditions had been specified, one gets a new surface onwhich stays, as a remnant of the operation of surgery just performed, a curve along whichthe field is constrained. This leads us to define marked surfaces.

Definition 1.3.1 A marked surface is a pair (M,C ), where M is a surface and C is afinite subset of pairwise disjoint smooth 1-dimensional submanifolds of the interior of M .The elements of C are oriented and we assume that each element of C appears in C withthe two possible orientations.

We denote by B(M) the set of connected components of the boundary of M , eachtaken with the two possible orientations. We denote by Conj(G) the set of conjugacyclasses of G. Observe that the inverse of a conjugacy class is still a conjugacy class.

1.3. MARKOVIAN HOLONOMY FIELDS 37

Figure 1.12: A surface marked by three curves.

Definition 1.3.2 By a set of G-constraints on (M,C ) we mean a mapping C : C ∪B(M) → Conj(G) such that for all N ∈ C ∪ B(M), and with the notation N−1 for thesame curve oriented in the opposite way, one has C(N−1) = C(N)−1.

Let (M,C ) be a marked surface and let l ∈ C be a mark. We denote by Spll(M) thesurface, not necessarily connected, which one obtains by cutting M along l. This surfaceis naturally endowed with a gluing map ψ : Spll(M) → M . The marks of M other thanl (resp. a measure of area on M , a set of G- constraints on (M,C )) determine marks onSpll(M), denoted by Spll(C ) (resp. a measure of area denoted by Spll(vol) on Spll(M), aset of G- constraints Spll(C) on (Spll(M), Spll(C ))).

Let (M,C , C) be a marked surface endowed with a set of G-constraints, let l ∈ C bea mark and let x be an element of G. We denote by Cl→x the set of G- constraints whichcoincides with C on all marks except l and l−1 and such that C(l) is the conjugacy classof x.

We can at last give the main definition.

Definition 1.3.3 (Markovian holonomy field [L7, Definition 3.1.2])Let G be a compact Lie group. A G-valued two-dimensional Markovian holonomy field

is the data, for each quadruplet (M, vol,C , C) consisting of a marked surface endowedwith a measure of area and a set of G-constraints, of a finite measure HFM,vol,C ,C on(M(P(M), G), I) such that the following properties are satisfied.

A1. For all (M, vol,C , C), HFM,vol,C ,C (∃l ∈ C ∪ B(M), h(l) /∈ C(l)) = 0.

A2. For all (M, vol,C ) and all event Γ ∈ I, the function C 7→ HFM,vol,C ,C(Γ) is ameasurable function of C.

A3. For all (M, vol,C , C) and all l ∈ C ,

HFM,vol,C−l,l−1,C|B(M)∪C\l,l−1=

∫G

HFM,vol,C ,Cl 7→x dx.

A4. If ψ : (M, vol,C , C) → (M ′, vol′,C ′, C ′) is a diffeomorphism which preserves allthe structures which we consider, then the mapping from M(P(M ′), G) to M(P(M), G)induced by ψ sends the measure HFM ′,vol′,C ′,C′ to the measure HFM,vol,C ,C.


A5. For all (M1, vol1,C1, C1) and (M2, vol2,C2, C2), one has the identity

HFM1∪M2,vol1∪vol2,C1∪C2,C1∪C2 = HFM1,vol1,C1,C1 ⊗ HFM2,vol2,C2,C2 .

A6. For all (M, vol,C , C) and all l ∈ C , denoting by ψ : Spll(M) → M the gluingmap along l, one has

HFSpll(M),Spll(vol),Spll(C ),Spll(C) = HFM,vol,C ,C ψ−1.

A7. For all (M, vol,∅, C) and for all l ∈ B(M),∫G

HFM,vol,∅,Cl 7→x(1) dx = 1.

The axioms A1, A2 and A3 are concerned with boundary conditions: they grant thefact that the measures HFM,vol,C ,C , as the marks and the constraints vary, are indeeddisintegrations of each other in the sense one expects them to be.

The axiom A4 ensures that the measure HFM,vol,C ,C does not depend on more structurethan what we have considered. Let us recall that a classical theorem of Moser asserts thatif two diffeomorphic surfaces are endowed with measures of area of the same total area,then there exists a diffeomorphism of the one onto the other which preserves areas. Thisresult can be refined in presence of marks (see Proposition 1.4.3 in [L7]).

The axiom A5 takes care of disjoint reunion of surfaces and allows one to restrictoneself to connected surfaces.

The axiom A6 is the Markov property itself. It encompasses all possible situationsregarding connectedness and orientability of the surface obtained after cutting M alongl. We mention orientability here because it can happen that a non- orientable surfacebecomes orientable when it is cut along a curve, for example a M obius band along anequator.

Finally, the equator A7 is a normalisation axiom. Without it, given a field HF, wecould for all real α 6= 0 define another field eαvol(M)HFM,vol,C ,C .

1.3.2 Existence and partial classification

We are now going to state our two main results which are a result of existence and aresult of classification. The natural order to state and prove these results is to start bythe classification result.

It is crucial to remember that the measures HFM,vol,C ,C are finite measures rather thanprobability measures. The transition kernels, which we have already discussed at length,are nothing but the masses of these measures.

Definition 1.3.4 Let HF be a Markovian holonomy field. Let p, g be non-negative integersand t a positive real. Let x1, . . . , xp be elements of G. Let M be an oriented surface withreduced genus g and whose boundary has p connected components. Let vol be a measure ofarea on M with total area t. Let finally C be the set of G-constraints along the boundary


of M which to the positively oriented and arbitrarily ordered components of the boundaryof M associates the conjugacy classes of x1, . . . , xp. We define

Z+p,g,t(x1, . . . , xp) = HFM,vol,∅,C(1).

Suppose now that g ≥ 1. Let M ′ be a non-orientable surface of reduced genus g and whoseboundary has p connected components. Let vol be a measure of area on M with total area t.Let C be a set of G-constraints along the boundary of M which to the arbitrarily orientedand arbitrarily ordered components of the boundary of M associates the conjugacy classesof x1, . . . , xp. We define

Z−p,g,t(x1, . . . , xp) = HFM ′,vol,∅,C(1).

The functions thus defined are called the partition functions of the holonomy field.

One proves that these functions satisfy the first part of Proposition 1.1.7 : they areall determined by the functions associated to disks and three-holed spheres.

The axiom A7 guarantees that each partition function is, with respect to each oneof its arguments and for all value of the others, the density of a probability measure onG. In particular, the family of measures (Z+

1,0,t(x) dx)t>0 is a one- parameter family ofprobability measures on G. In order to study it, and more generally in order to studythe holonomy fields which we have just defined, it is necessary to make a few regularityassumptions.

Definition 1.3.5 Let HF be a Markovian holonomy field. We say that HF is regular ifthe following two conditions are satisfied.

1. For all (M, vol,C , C), and for all sequence (cn)n≥0 of paths on M which convergeswith fixed endpoints to c, one has∫

M(P(M),G)

dG(h(cn), h(c)) HFM,vol,C ,C(dh) −→n→∞

0.

2. For all (M, vol,C ), the function (t, C) 7→ HFM,tvol,C ,C(1) is continuous.

For a regular holonomy field, the second part of Proposition 1.1.7 is true : all partitionfunctions are determined by the Z+

1,0,t. The next theorem clarifies their structure andclassifies partially the random holonomy fields.

Theorem 1.3.6 ([L7, Proposition 4.2.1]) Let HF be a regular Markovian holonomyfield. Then the probability measures (Z+

1,0,t(x) dx)t>0 on G are the 1-dimensional marginaldistributions of a conjugation invariant Levy process issued from the unit element. Thisprocess is said to be associated to HF and determines all its partition functions. It isadmissible in the sense of Definition 1.2.4.

It is tempting to think that this Levy process determines completely the holonomyfield, but, except when G is Abelian, this does not seem easy to prove. On the otherhand, the construction which we have explained in the previous sections allows one toprove the next theorem.


Theorem 1.3.7 ([L7, Theorem 4.3.1]) Every admissible Levy process on a compactLie group is associated with a regular Markovian holonomy field.

This theorem is probably the most important in this whole chapter and the mostsignificant result in my attempts to improve and generalise the construction of the Yang-Mills measure. Its proof occupies most of the monograph [L7].

1.3.3 The Yang-Mills field

Among all admissible Levy processes on a connected compact Lie group, the Brownianmotion plays a special role. The holonomy field constructed by applying Theorem 1.3.7to the Brownian motion has a name: it is the Yang-Mills field. It is the first to have beenconstructed and it is the one which physicists use in quantum field theories. We will comeback to this particular field in the next chapter where we will investigate, in the form of alarge deviation principle, the links between the object that we have constructed and theexpressions which physicists manipulate with such great success.

In this section, we give a simple description of this field, analogous to that of thePoisson process indexed by loops (see Section 1.1.1), when the group G is Abelian, thatis, since we also suppose it compact and connected, when G = U(1)n for some n ≥ 0. Thiscase has been the first to be understood and the first works devoted to the constructionof the Yang-Mills field with more general groups have proceeded by reducing the problemto the Abelian case. For the sake of simplicity, we treat the case n = 1.

Let us work, to start with, in the plane R2. Let W be a white noise on R2, that is,an isometry of L2(R2) into a linear space of centred Gaussian random variables. Equiv-alently, W is the Gaussian process indexed by L2(R2) whose covariance is given by theL2 scalar product. We will denote by W (f) the Gaussian variable associated with asquare-integrable function f on R2.

By analogy with Definition 1.1.1, we associate to each loop l the random variable

Bl = eiW (nl). (1.8)

This definition makes sense only if nl is square-integrable. We have discussed this pointearlier and indicated that the Banchoff-Pohl inequality guarantees us that this is the casewhen l has finite length (see (1.7) and the discussion which precedes it).

If we restrict ourselves to a disk D contained in R2, the family of variables (Bl)l∈L(D)

thus defined has the distribution of (Hl)l∈L(D) under the measure∫U(1)

HFD,vol,∅,∂D 7→x dx,

where HF is the Yang-Mills field which we integrate here with respect to the boundarycondition. According to the axiom A7, this measure is indeed a probability measure.

In order to put a specific boundary condition on the boundary of the disk D, let usintroduce a variant of the function nl by setting

n0l = nl −

1

vol(D)

∫D

nl dvol,

that is by subtracting its mean to nl. If we define, for each rectifiable loop l in D,B0l = eiW (n0l ), then we obtain a process indexed by L(D) which resembles the process B


and satisfies B0∂D = 1 almost surely. However, the distribution of (B0

l )l∈L(D) is not equal tothat of the canonical process under the normalised measure Z+

1,0,vol(D)(1)−1HFD,vol,∅,∂D 7→1.

In order to see this, let us make again in this context the construction which gave rise tothe process (Nlt)t≥0 in Proposition 1.1.3. Suppose to fix the ideas that D is the disk ofradius 1√

πcentred at the origin and let us call, for all t ∈ [0, 1], lt the circle centred at the

origin and with radius√t√π. Then the function n0

ltequals 1 − t in the disk bounded by lt

and −t outside. It follows immediately that for all s, t ∈ [0, 1],

〈n0ls , n

0lt〉L2(D,vol) = min(s, t)− st.

Thus, the real process (W (n0lt))t∈[0,1] is a Brownian bridge from 0 to 0 of length 1, and

the process (B0lt)t∈[0,1] is the exponential of (i times) this bridge. But this process is not

Markovian. Indeed, its trajectories are all homotopically trivial in the circle U(1) : it isthe bridge from 1 to 1 of length 1 of the Brownian motion on U(1) conditioned by the factthat its trajectories are homotopic to the constant loop. This conditioning is not singularbut it destroys the Markov property.

The bridge of length 1 of the Brownian motion on U(1) is the exponential of i timesthe Brownian motion on R conditioned not to come back to 0, but rather to arrive at apoint of 2πZ at time 1. Its final value is thus a random variable T whose distribution isgiven by

∀k ∈ Z, P(T = 2kπ) ∝ e−2k2π2

. (1.9)

A better definition consists then in taking a random variable T whose distribution is thatdescribed above, independent of the white noise W , to set nl = 1

vol(D)

∫Dnl dvol, and then

B(1)l = ei(W (n0l )+nlT).

The process B1 thus defined has the distribution of the canonical process under themeasure Z+

1,0,vol(D)(1)−1HFD,vol,∅,∂D 7→1.

In order to define the field on more general surfaces, it is necessary to extend thedefinition of the function n0

l and the number nl. Such an extension is possible, at the priceof certain choices. Let M be an oriented surface with reduced genus g. Let N1, . . . , Np

denote the boundary components of M , positively oriented. Let C1, . . . , Cg be a familyof smooth oriented curves on M (it suffices that they be rectifiable) which determine abasis of the first homology group of the surface obtained from M by gluing a disk alongeach boundary component. Then the classes of N1, . . . , Np, C1, . . . , Cg generate H1(M,Z)with the relation [N1] + . . . + [Np] = 0. Let now l be a loop on M . Its homology classdecomposes uniquely as

[l] =

g∑i=1

ζi[Ci] +

p−1∑j=1

νj[Nj],

where the ζi and the νj are integers. The cycle

l⊥ = l −g∑i=1

ζiCi −p−1∑j=1

νjNj


is thus homologous to 0. There exists then a unique function on M with zero mean,locally constant on the complement of the range of l⊥ and which varies by 1 or −1 ateach crossing of l⊥, depending on the orientation of M and l⊥. This definition is of courseonly valid if l⊥ is regular enough but we will not enter the detail of the definition. Wewill denote this function by n0

l⊥ . The values that it takes are all equal modulo 1 and wedenote by −nl the corresponding class of R/Z. One checks easily that this notation isconsistent with the case of the disk studied above.

We can now define a process indexed by the loops on an arbitrary surface. We use thenotation of the paragraph above.

Definition 1.3.8 Let (M, vol,∅, C) be a surface endowed with a measure of area andboundary conditions. Let x1, . . . , xp denote the boundary conditions along N1, . . . , Np. Setx = x1 . . . xp. Let W be a white noise on (M, vol), that is, an isometry from L2(M, vol)into a real Gaussian space. Let T be a random variable with values in t ∈ R : eit = x,independent of W and with distribution

∀t ∈ R, eit = x, P(T = t) ∝ e−t2

2vol(M) .

Let U1, . . . , Ug be uniform independent variables on U(1), independent of W and T . Forall loop l ∈ L(M), we set

B(x1,...,xp)l = eiW (n0

l⊥)einlTU ζ1

1 . . . U ζgg x

ν11 . . . xνpp . (1.10)

Theorem 1.3.9 ([L2, Theorem 3.2]) The process B(x1,...,xp) thus defined has the samedistribution as the canonical process under the normalised measure Z+

p,g,vol(M)(x1, . . . , xp)−1

HFM,vol,∅,C.

In the Abelian case, the structure of the holonomy field associated to the Brownianmotion is thus that of a white noise to which a finite-dimensional alea is added. It is forinstance possible to reconstruct the white noise starting from the random holonomy field.This indicates, incidentally, that the measure of area is entirely encoded in the holonomyfield.

In the next chapter, we will discuss the interest and significance of the definition (1.10)when one substitutes in it to the variable T a deterministic value t such that eit = x.

1.3.4 Link with topological quantum field theories

We have presented the definition of random holonomy fields (Definition 1.3.3) as a gener-alisation of the definition of usual Markov processes, indexed by intervals of time. In thissection, we are going to emphasise another analogy, between this definition and that of atopological quantum field theory (abbreviated TQFT), due to M. Atiyah.

INSERT 7 – Topological quantum field theories [Ati88]

Let n ≥ 0 be an integer. There is a category, denoted by Cobn, whose objects are the compact orientedsmooth manifolds without boundary of dimension n and whose morphisms are given by the cobordisms: ifN1 and N2 are two oriented n-manifolds, then Hom(N1, N2) is the set of oriented (n + 1)- manifolds the


boundary of which is endowed with an orientation-preserving diffeomorphism with the disjoint union of N∗1and N2, where N∗1 denotes the manifold obtained from N1 by reversing the orientation. On the other hand,there exists a perhaps much more familiar category, denoted by Vect, whose objects are complex linear spacesand whose morphisms are linear mappings.

A (n+ 1)-dimensional TQFT is a covariant functor from Cobn into Vect. This functor is usually denotedby Z, in reference to the fact that this definition was initially given as a general framework for the partitionfunctions of certain statistical or quantum field theories. Thus, by definition, Z associates to each n-manifoldN a linear space Z(N) and to each (n+1)-manifold M endowed with a diffeomorphism between its boundaryand the disjoint union N∗1 tN2 of two n-manifolds, a linear map Z(M) : Z(N1)→ Z(N2).

n = 0 n = 1

Figure 1.13: Two examples of cobordisms, in dimensions n+ 1 = 1 and n+ 1 = 2.

This functor is required to satisfy a certain number of natural properties.1. To be multiplicative: for all n-manifolds N1 and N2, we insist that Z(N1 t N2) = Z(N1) ⊗ Z(N2)

(the tensor product of linear spaces) and, for any two (n + 1)-manifolds M1 and M2, that Z(M1 tM2) =Z(M1)⊗ Z(M2) (the tensor product of linear maps).

2. To be involutive: for all n-manifold N , we insist that Z(N∗) = Z(N)∗ (the dual linear space) and, forall (n+ 1)-manifold, that Z(M∗) = Z(M)∗ (the adjoint linear map).

3. To behave well with respect to the composition of cobordisms: if M1 realises a cobordism between N1

and N2, and M2 a cobordism between N2 and N3, and if M is formed by gluing M1 and M2 along N2, thenwe insist that Z(M) = Z(M2) Z(M1).

Z(S1)⊗4 Z(M1)−→ Z(S1)⊗5 Z(M2)−→ Z(S1)⊗2

M1 M2

Figure 1.14: Composition of two cobordisms in dimension 2.

This last axiom can be reformulated in a more general way by slightly shifting our point of view. If weidentify the space of linear maps from Z(N1) into Z(N2) with Z(N1)∗ ⊗ Z(N2) = Z(N∗1 tN2), we can say


that the linear map Z(M) is an element of the linear space Z(∂M). In these terms, the axiom becomes thefollowing.

3’. If the boundary of M can be written as N1 tN∗1 tN2 and if M ′ is formed by gluing the two copiesof N1, then we insist that Z(M ′) = κ(Z(M)), where κ : Z(N1)∗ ⊗ Z(N1)⊗ Z(N2)→ Z(N2) is the naturalcontraction of the first two factors.

M M ′

N1

N∗1

N2 N2

Figure 1.15: Gluing two components of the boundary of a surface.

4. Finally, we insist that Z behaves well with respect to orientation-preserving diffeomorphisms of man-ifolds of dimension n and n + 1, that is to say that to diffeomorphic n-manifolds it associates isomorphiclinear spaces and to diffeomorphic cobordisms it associates linear maps conjugated by the linear isomorphismsassociated to the diffeomorphisms induced on the boundaries.

If ∅ designates the empty n-manifold, then the multiplicativity condition implies for example that Z(∅) =Z(∅) ⊗ Z(∅). Thus, Z(∅) is either the null linear space or C. In order to avoid trivial situations, we willassume that Z(∅) = C. Thus, for all (n + 1)-manifold M without boundary, Z(M) is a linear map from Cinto itself, that is, a complex number.

If N is an n-manifold, then Z(N × [0, 1]) is an endomorphism of Z(N). Since the cylinder N × [0, 1] isdiffeomorphic to two copies of itself glued together along two copies of N , one has the identity

Z(N × [0, 1]) = Z(N × [0, 2]) = Z(N × [0, 1]) Z(N × [0, 1]).

Thus, Z(N × [0, 1]) is a projection in the linear space Z(N) and, since a surface is not modified up tohomeomorphism when one glues a cylinder along one of its boundary components, one loses no generality inassuming that it is the identity. Then, gluing the two ends of this cylinder and using the axiom 3′, on findsthat Z(N × S1), where S1 is the circle, is the natural contraction of the identity of Z(N), that is, the traceof this linear map, which is nothing but the dimension of Z(N).

In the context of Markovian holonomy fields, we should modify slightly the definitionof a TQFT in order to incorporate in it the notion of volume, or area when n + 1 = 2.We just need to replace (n+ 1)-manifolds by manifolds endowed with a notion of (n+ 1)-volume, of which the only invariant under diffeomorphisms is, according to a theorem ofMoser, the total volume. In doing this, we modify only the morphisms in the categoryCobn. The axioms 1 to 4 are unchanged. The object thus defined is called an area-dependent topological quantum field theory (ad-TQFT). For all (n + 1)- manifold M , wewill denote by Zt(M) the element of Z(∂M) associated to M when it is endowed with ameasure of (n+ 1)-volume of total area t.

Let us investigate a 1-dimensional ad-TQFT, which we denote by Z. There is only oneconnected 0-manifold, the point, which we denote by pt. It can have two orientations,denoted by + and −. There is only one linear space involved, namely Z(pt+), which


we denote by V . Then, there are only two oriented connected 1- manifolds, the intervaland the circle. It is convenient to represent intervals as intervals of R endowed with theirnatural length. Thus, for all s < t reals, Z([s, t]) is an endomorphism of V , which dependsonly on the length of [s, t] and which we denote by Pt−s. Moreover, the axiom 3 impliesthat the relation Z([s, u]) = Z([t, u]) Z([s, t]) holds for all s < t < u, so that (Pt)t>0 is asemigroup of endomorphisms of V . Finally, according to the general argument which wehave presented above, we have for all t > 0 the relation Zt(S

1) = Tr(Pt).Finally, a 1-dimensional ad-TQFT is the same thing as a semigroup of endomorphisms

of a linear space. In contrast with the purely topological case, nothing here prevents thislinear space from being infinite-dimensional. For instance, in a Hilbertian framework, itsuffices that the semigroup should be of trace class.

A sufficiently regular classical Markov process, indexed by intervals of time, gives rise,through its transition kernels, to a 1-dimensional ad-TQFT. Consider for instance theBrownian motion X on a compact Lie group G. For all t > 0, let Qt denote the density ofthe distribution of Xt with respect to the Haar measure on G. Define Z(pt) as the spaceL2(G)G of square-integrable functions on G which moreover are invariant by conjugation.Then, for all s < t, we can define Z([s, t]) as the mapping (x, y) 7→ Qt−s(x

−1y), seeneither as the integral kernel of a linear map from L2(G)G into itself, or equivalently as an

element of L2(G2)G2 ' L2(G)G ⊗ L2(G)G.

Our definition of the 2-dimensional Markovian holonomy fields contains the definitionof an ad-TQFT, given by the partition functions of the field. In the 2-dimensional setting,there is also a unique connected 1-manifold, which is the circle S1, so that there is alsoa unique linear space involved. We set Z(S1) = L2(G)G. Note that this space is finite-dimensional when G is finite. Then the partition functions Z±p,g,t of a Markovian holonomyfield, which are indeed symmetric, invariant by conjugation and square-integrable withrespect to all their arguments, can thus be seen as a symmetric tensor of (L2(G)G)⊗p. Letus summarise this whole section in the following result.

Proposition 1.3.10 Let G be a compact Lie group. Let (Z+p,g,t)p,g≥0,t>0 be the partition

functions of a Markovian holonomy field, associated to oriented surfaces. Set Z(S1) =L2(G)G, the space of square-integrable functions on G invariant by conjugation. For allsurface M with genus g, total area t and such that ∂M has p connected components, setZ(M) = Z+

p,g,t. Then the functor Z is a 2-dimensional ad-TQFT.

Chapter 2

Holonomy fields, coverings andconnections

In this second chapter, we are going to describe several geometrical structures which giverise to multiplicative functions on the set of loops on a surface, with values in a group. Intwo very different situations, we are going to prove that some of the processes constructedin the previous chapter can be related to random versions of these geometrical structures.

2.1 Finite groups and ramified coverings [L7, Chapter 5]

In this section, we are going to explain, when the group G is finite, how the Markovianholonomy field associated by Theorem 1.3.7 to a random walk on G can be realisedexplicitly by random ramified coverings of the surface on which one is working. Presentingthis result will also give us the opportunity to understand in a simple situation how randomholonomy fields are related to notions of horizontal lift and parallel transport.

2.1.1 Fundamental group and coverings

We are going to start by looking for the simplest possible geometric structures which giverise to multiplicative functions on the set of loops on a surface, with values in an arbitrarygroup.

Let M be a surface, on which we fix a point m. Each loop l ∈ Lm(M) has a homotopyclass, which is an element of the fundamental group π1(M,m). The mapping l 7→ [l] whichto each loop associates its homotopy class is a multiplicative function Lm(M)→ π1(M,m),which is given by the bare topological structure of M . We are going to reinterpret thismapping in a way which apparently is rather complicated but will prove helpful in under-standing many more multiplicative functions.

Let π : M → M be a universal covering of M . Choose m ∈ π−1(m). The choice

of m determines an action of the group π1(M,m) on M , in the following way. Let [l]

be a homotopy class of loops based at m, represented by a loop l. Let l be the lift ofl issued from m and let n be the final point of l, which does not depend on the choice

of l in [l]. There exists a unique automorphism of the covering π : M → M (that is, a

47

48 CHAPTER 2. HOLONOMY FIELDS, COVERINGS AND CONNECTIONS

homeomorphism h of M such that π h = π) which sends m on n. This automorphismof covering is the one which we associate to the element [l] of π1(M).

We have thus a covering of M (here π : M →M) and a group (here π1(M,m)) whichacts simply and transitively on the fibres of this covering, which means that for each pairof elements of the same fibre, there exists a unique element of the group which sends thefirst on the second. We have moreover a basepoint on M (here m) and an element of thecovering above this basepoint (here, m). Given a loop l on M based at m, we can lift thisloop starting from m, determine the final point n of this lift and associate to the loop lthe unique element of the group which sends m on n. Here, tautologically, this elementis the homotopy class of l.

In order to understand what we have gained with this obscure formulation, let uschoose a group G, for example a finite group, and a homomorphism of groups ρ :π1(M,m) → G. Then we can form the multiplicative function l 7→ ρ([l]) from Lm(M)into G. This multiplicative function can be interpreted just in the same way as in thelast paragraph. After replacing G by one of its subgroups if necessary, we will assumethat ρ is onto. The theory of coverings guarantees us that to the kernel of ρ, which isa normal subgroup of π1(M,m), is associated a covering E → M whose automorphismgroup identifies with the quotient of π1(M,m) by the kernel of ρ, which in turns identifies

with G. A way of constructing E is to realise it as the fibred product M×π1(M,m)G, which

is the quotient of the Cartesian product M ×G by the equivalence relation which for all

x ∈ M , all g ∈ G and all γ ∈ π1(M,m), identifies (x, g) and (x · γ, ρ(γ−1)g).This way of constructing E shows thatG acts on E on the right, simply and transitively

on the fibres: it suffices to set (x, g) · g′ = (x, gg′). The covering E is thus a principalG-bundle.

Definition 2.1.1 Let G be a finite group. By a principal G-bundle over M we mean acovering π : E → M endowed with an action on the right of G on E, which is free andtransitive on the fibres.

The construction which we have already done twice is the following one.

Proposition 2.1.2 Let G be a finite group. Let π : E →M be a principal G-bundle. Letm be a point of M . Let e be a point of E such that π(e) = m. Let l be a loop on M based

at m. Let l be the horizontal lift of l issued from e. The unique element of G which sendse on the final point of l is called the monodromy or holonomy of l. The function fromLm(M) to G which to a loop associates its holonomy is multiplicative.

This proposition is the fundamental example, and the simplest one, of the way in whichwe are going to interpret multiplicative functions in geometric terms. It is however not avery rich source of multiplicative functions. On one hand, the only groups G for whichone can make a non-trivial construction are those which are isomorphic to quotients of thefundamental group of M . On the other hand, the image of a loop by the multiplicativefunctions which we have constructed so far depends only on its homotopy class. If M issimply connected, for example a sphere or a disk, we can only construct the multiplicativefunction which is identically equal to the unit element. If the fundamental group of M is

2.1. FINITE GROUPS AND RAMIFIED COVERINGS [L7, CHAPTER 5] 49

Abelian, for example if M is a torus or a projective plane, then G must be Abelian. Inthe case of the projective plane, it can actually only be Z/2Z.

2.1.2 Random ramified coverings

In order to enrich this construction, we are going to allow ourselves to puncture M , thatis, to remove a finite set of points from it. This will solve our problems, since on one handthe fundamental group of a surface of which one has removed a finite non-zero numberof points is always a free group, and on the other hand removing points will allow usto distinguish between loops which were originally homotopic in M . The drawback is ofcourse that we cannot consider anymore those loops which cross the finite set which wehave removed. However, this set is meant to become a random subset of M , and theprobability that a given loop meets this set will be zero.

Definition 2.1.3 A ramified principal G-bundle on M with ramification locus Y , whereY is a finite subset of the interior of M , is a covering π : R → M \ Y endowed with anaction of G on the right on R, which is free and transitive on the fibres.

The structure of a ramified bundle in the neighbourhood of a ramification point issimple. If y is a point of the ramification locus Y , there exists a neighbourhood V of ysuch that the restriction of π to each connected component of π−1(V \y) is topologicallyconjugated to the mapping z 7→ zn from C∗ into itself, for a certain integer n ≥ 1 (seeFigure 2.1). In particular, it is possible to add a finite number of points to R and tomake it a smooth surface, in such a way that all the new points are sent on Y by π. Themapping π : R→M thus obtained is called a ramified covering.

0

1

1

10

−1

−1

i

−i

z 7→ z2

Figure 2.1: The mapping z 7→ z2 is the simplest ramified covering of C by itself.

The conjugacy class of the monodromy along a small circle around a ramification pointy does not depend on the way in which it is computed. It is called the local monodromy


at y and we denote it by O(R, y). We will use the name ramification point only for thosepoints where the local monodromy is not the unit element.

Let π : R→M be a ramified bundle. Let N be a connected component of the bound-ary of M . The restriction of π to π−1(N) is a principal G-bundle over N , without anysingularity since we have assumed that all ramification points are contained in the interiorof M . The structure of this G-bundle is completely determined, up to isomorphism, bythe monodromy along N . In fact, again, only the conjugacy class of this monodromy iswell-defined, unless one chooses a basepoint on N and a point in the fibre over this point.

We are going to sort the ramified G-bundles according to their ramification locus andtheir structure over the boundary. We consider in fact isomorphism classes of G- bundles,where an isomorphism between π : R → M and π′ : R′ → M is a diffeomorphismψ : R → R′ which intertwines the actions of G on R and R′ and which is such thatπ′ ψ = π.

Definition 2.1.4 Let M be a surface. Let C : B(M)→ Conj(G) be a set of G-constraintson the boundary of M (see Definition 1.3.2). Let Y be a finite subset of the interior ofM . We denote by R(M,Y,C) the set of isomorphism classes of ramified G-bundles overM with ramification locus Y and boundary structure given by C. We denote by R(M,C)the union of the sets R(M,Y,C) as Y spans the set of finite subsets of the interior of M .

We will denote by O1, . . . ,Op the p conjugacy classes of G associated by C to the pconnected components of the boundary of M .

We are now going to define a probability measure onR(M,C). Let us start by choosingan admissible Levy process X on G (see Definition 1.2.4), characterised by its jumpmeasure Π, which is a measure on G \ 1. We denote by Π1 the probability measureΠ/Π(G). The definition of the measure is done in three steps.

1. Let R be a ramified G-bundle with ramification locus Y . We call weight of R thefollowing non-negative real:

Π(R) =∏y∈Y

Π1(O(R, y))

#O(R, y).

It is the product over all ramification points of the mass attributed by the proba-bility measure Π1 to an arbitrary element of the conjugacy class which is the localmonodromy at this point.

2. One defines a finite measure RBXM,Y,C on R(M,Y,C) by setting

RBXM,Y,C =#G1−g

#O1 . . .#Op∑

R∈R(M,Y,C)

Π1(R)

#Aut(R)δR.

It is essentially the uniform measure on the finite set R(M,Y,C), with a weightingdetermined by the monodromy and our choice of X, and the fact that, as oftenin combinatorics, objects with many symmetries count less than objects with fewsymmetries.

2.2. BROWNIAN HOLONOMY FIELD AND RANDOM CONNECTIONS [L4, LN5] 51

3. Let F(M) denote the set of finite subsets of M , endowed with a natural topology. Weconsider on F(M) the probability measure Ξ which is the distribution of a Poissonpoint process with intensity vol. We define a finite measure RBXM,vol,C on R(M,C)by

RBXM,vol,C =

∫F(M)

RBXM,Y,C Ξ(dY ).

For some subsets Y ⊂ M , the set R(M,Y,C) can be empty. It is for example thecase if M has no boundary and G = Z/2Z : if Y has an odd cardinal, then R(M,Y,C)is empty, whereas if Y has even cardinal, R(Y,M,C) contains a unique element whichadmits exactly two automorphisms: the identity and the exchange of the two sheets. Inthis case, the ramification locus of a covering picked under the measure RBXM,vol,C is notexactly a Poisson point process, but rather a Poisson process conditioned to have an evennumber of points. In all cases, the distribution of the ramification locus of this randomcovering is absolutely continuous with respect to Ξ.

Let l1, . . . , ln be loops on M based at the same point m. With Ξ- probability 1, theset Y meets the range of none of these loops. Let f : Gn → R be a continuous functioninvariant by the diagonal action of G by conjugation. If we pick a ramified G-bundle atrandom under the measure RBXM,vol,C (normalised) and if we choose a reference point in thefibre above m, we can compute the monodromies h(l1), . . . , h(ln) of the loops l1, . . . , ln. Ifwe change the reference point, each monodromy is conjugated by the same element of G.Thus, f(h(l1), . . . , h(ln)) is well-defined, independently of the choice of a reference point.Considering the definition of the σ-field I (Definition 1.2.8), this makes it plausible thatthe measure RBXM,vol,C induces a finite measure on (M(P(M), G), I), which we denote by

MFXM,vol,C . The main result is then the following.

Theorem 2.1.5 ([L7, Theorem 5.4.2]) The finite measures HFXM,vol,∅,C and MFXM,vol,C,defined on the space (M(P(M), G), I), are equal.

We have thus reached our goal and realised the Markovian holonomy field associated toa random walk on a finite group as the field of random monodromy of a random ramifiedcovering of a surface.

2.2 Brownian holonomy field and random connections [L4, LN5]

We are now going to interpret certain holonomy fields with values in continuous groupsin terms of analogous structures to those which we have just described. Unfortunately,the geometrical objects are somewhat more complicated and there is no such satisfactoryresult as Theorem 2.1.5 in this case. We are going to focus on the field associated tothe Brownian motion on a compact connected Lie group. For this field, we are going towrite a large deviation principle which relates the process which we have constructed toan energy functional of geometric origin, namely the Yang-Mills action.


2.2.1 A random differential 1-form

Let us start by examining the case where the group G is Abelian. Let M be a surface.When G is Abelian, say G = U(1), there is a natural function to produce multiplicativefunctions of loops, by choosing a differential 1-form α on M and by associating to eachloop l the element exp i

∫lα of U(1). We are going to prove that the Brownian motion on

U(1) indexed by loops on a disk, which we have described in Section 1.3.3 and defined bythe equation (1.8), can be understood heuristically as the exponential of the integral of arandom differential 1- form.

Let us work on the plane or on a disk. Let us consider a loop l. Disregarding anyconsideration of rigour, we treat the white noise W as a random function, so that it is forinstance possible to write W (nl) =

∫R2 nl(x)W (x) dx (see Sections 1.1.1 and 1.3.3). We

want to write this last integral as the integral of a 1-form along l. To start with, let usdefine, for all x = (x1, x2) ∈ R2, the 1- form dθx on R2 \ x by setting

∀y = (y1, y2) ∈ R2, dθx(y) =(y2 − x2)dy1 − (y1 − x1)dy2

(y1 − x1)2 + (y2 − x2)2.

The index of l with respect to x can be evaluated thanks to the form dθx, by theformula

nl(x) =1

2π

∫l

dθx.

Let us introduce the function G(x, y) = − 12π

log d(x, y), where d is the Euclidian distance.This function is defined outside the diagonal on R2 × R2 it is the kernel of the inverse ofthe Laplace operator. This means that if, for each compactly supported smooth functionψ we set Gψ(x) =

∫R2 G(x, y)ψ(y) dy, then we have, for all compactly supported smooth

function ϕ : R2 → R, the equality G∆ϕ = ϕ.For all x, let us denote by Gx the function G(x, ·) on R2 \ x. The differential of this

function Gx resembles closely dθx. In fact, we have

1

2πdθx = ∗dGx,

where ∗ is the Hodge operator defined by ∗(a dy1 + b dy2) = −b dy1 + a dy2, so thatdifferential 1-form α and all vector X, one has (∗α)(X) = α(J−1X), where J denotes therotation of angle π

2.

Combining formally these observations, we find

W (nl) =

∫R2

nl(x)W (x) dx

=

∫R2

∫l

∗dGxW (x) dx

=

∫l

∗dy∫R2

G(x, y)W (x) dx

=

∫l

∗d(GW ).


We have thus heuristically defined a random 1-form by taking a white noise, applying toit the inverse of the Laplace operator, taking the total differential of the function thusobtained, and applying the Hodge operator. The exponential of the integral of this 1-form gives the Brownian motion indexed by loops.

Since we integrate it along loops only, we have the freedom to modify this 1-formby arbitrarily adding to it an exact 1-form. This corresponds to the invariance of theobservable quantities under the action of the gauge group. We are now going to describethe appropriate geometric object which, without the assumption that G is Abelian, allowsone to define multiplicative functions of loops.

2.2.2 Connections

Let us start by defining the object which, in the case of a continuous group, plays the roleof a ramified covering.

Definition 2.2.1 Let M be a surface. Let G be a compact Lie group. A principal G-bundle on M is a smooth manifold P endowed with a smooth mapping π : P →M and afree action of G, denoted by (p, g) 7→ pg, such that π induces a diffeomorphism from thequotient P/G onto M .

So, P is the disjoint union of the fibres π−1(m) where m runs over M and, for eachm, the group G acts freely and transitively on π−1(m). This definition is very similar tothe definition which we have given in the finite case. However, in the finite case, the totalspace of the bundle, which we have denote by E and R, was a surface itself, whereas inthe present case, it is a manifold whose dimension is the sum of that of M and that of G.

Definition 2.2.2 Let π : P → M and π′ : P ′ → M be two principal G-bundles over M .An isomorphism of P on P ′ is a diffeomorphism ϕ : P → P ′ such that π′ ϕ = π andwhich commutes to the actions of G on P and P ′, that is, such that for all p ∈ P and allg ∈ G, one has ϕ(p · g) = ϕ(p) · g.

The group of automorphisms of a bundle π : P →M is called the gauge group of thisbundle. We denote it by J (P ), or simply by J .

The fundamental example of a principal G-bundle over M is the Cartesian productM ×G, on which G acts according to the formula (m,h) · g = (m,hg). The gauge groupof M ×G, instead of being a finite group as in the case of ramified bundles, is now a hugegroup, which identifies naturally with the group of all smooth mappings from M to G. Ifj : M → G is such a mapping, then it acts on P according to j · (m, g) = (m, j(m)g).

The local structure of a principal bundle is always that of a Cartesian product : therestriction of π : P → M to π−1(U), where U is a small open subset of M , is alwaysisomorphic to the trivial bundle U × G → U . Nevertheless, there exist bundles whoseglobal structure is not that of a Cartesian product, as for example the covering of C∗ byitself by the mapping z 7→ z2, which is a Z/2Z- principal bundle, but not isomorphic toC∗ × Z/2Z→ C∗.

A fundamental difference between the discrete and continuous cases is that, in a bundlewhose structure group is continuous, there is no canonical way of lifting a path in M to


a path in P . A choice is necessary, namely that of a connection on P . A connectionallows one to lift a tangent vector to M to a tangent vector to P . By solving differentialequations, this allows one to lift piecewise differentiable paths.

More precisely, a connection defines (see Figure 2.2), for all m ∈M and all p ∈ π−1(m),an injective linear mapping rpm : TmM → TpP , which on one hand is a horizontal lift1,in the sense that, when it is composed with the differential of π, it gives the identity:Tpπ rpm = idTmM ; and which on the other hand behaves well with respect to the actionof G: for all g ∈ G, denoting by Rg the action of g on the right on P and TpRg its

differential at p, we have rpgm = rRg(p)m = TpRg rpm.

M

P

m

TmM

π−1(m)

p

pg

TpP

TpgP

TpRgrpgm

rpmTpgπ

Tpπ

π

Figure 2.2: This picture illustrates the compatibility relations between the mappings of horizontal liftrpm, the projection π and the action of G on P . The two shaded subspaces of TpP and TpgP are thehorizontal spaces at p and pg.

1The french word for horizontal lift is relevement horizontal, hence the choice of the letter r.


In order to describe a connection, one generally describes the range of the mapping rpmat each point of P . It is a subspace of the dimension of M in the tangent space to P at p.This subspace admits a canonical supplementary subspace, which is the tangent space tothe fibre through p. We denote this subspace, called the vertical subspace, by T Vp P ⊂ TpP .It is canonically identified with the Lie algebra g of G by the differential at the unit elementof the action of G: each vector of T Vp P can be uniquely written as d

dt |t=0petA for some A in

g. One describes a connection by specifying, at each point p ∈ P , the projection on T Vp P

along the horizontal subspace. With the identification T Vp P ' g, this projection becomesa linear mapping TpP → g, whose kernel is the horizontal subspace.

p

TVp P = Tp(π

−1(m))

THp P

π−1(m)

1 ∈ G

G

g = T1G

g 7→ p · g

θp : A 7→ ddt |t=0

p · etA

X

XH

XV

ω(X) = θ−1p (XV )

Figure 2.3: The connection 1-form is the projection on the vertical space along the horizontal space,seen as taking its values in the Lie algebra of G thanks to the identification θp.

The technical definition is the following2.

Definition 2.2.3 Let π : P → M be a principal G-bundle over M . A connection on Pis a differential 1-form ω on P with values in g which satisfies the following properties.1. For all A ∈ g, all p ∈ P , one has ω( d

dt |t=0petA) = A.

2. For all p ∈ P , all X ∈ TpP , all g ∈ G, one has ω(TpRg(X)) = ad(g−1)ω(X) =g−1ω(X)g.

We denote by A(P ), or simply A, the space of connections on P .

The gauge group J (P ) acts on the space A(P ) of connections on P , the gauge trans-formation j sending the connection ω to the connection j∗ω defined by (j∗ω)p(X) =ωj(p)(Tpj(X)). The space A(P ) is an infinite dimensional affine space and J (P ) acts by

2For more details on principal bundles and connections, one can consult the first volume of the book byKobayashi and Nomizu [KN96] and the book by Morita [Mor01].


affine transformations. The quotient space A(P )/J (P ) is the configuration space of thephysical theory which motivates the study of the Yang-Mills field.

Given a connection ω ∈ A(P ), a path c : [0, 1] → M with c(0) = m and a pointp ∈ π−1(m), there exists a unique path c : [0, 1] → P such that π(c(t)) = c(t) for allt ∈ [0, 1] and ω( ˙c(t)) = 0 for all t ∈ [0, 1]. The path c is called the horizontal lift of cstarting at p (see Figure 2.4). If c is a loop, that is, if c(0) = c(1), the point c(1) belongsto π−1(m) and can be written as p · h for a unique h ∈ G. This element h is calledthe holonomy of ω along c, computed at p. We denote it by holp(ω, h). The invarianceproperties of the connection with respect to the action of G on P imply that for all g ∈ G,the horizontal lift of c issued from p · g finishes at c(1) · g = p · hg = (p · g) · g−1hg. Thus,we find the relation

holpg(ω, c) = g−1holp(ω, c)g.

From this relation follows the fact that if l1 and l2 are two loops based at m and if l1l2denotes their concatenation, then

holp(ω, l1l2) = holp(ω, l1)holpholp(ω,l1)(ω, l2) = holp(ω, l2)holp(ω, l1).

This justifies the order which we have chosen in the definition of multiplicativity (Defini-tion 1.2.1).

M

π−1(c([0, 1]))

c

p q

p · g q · g

M

l

pp · h

p · gp · hg

l

m

π−1(m)

Figure 2.4: Each horizontal lift of a path is an integral curve of a vector field defined in the union ofthe fibres over the range of the path. This field is obtained by taking all horizontal lifts of the speeds ofthe path. If the path is a loop, it is possible to compare the finishing point of the horizontal lift with itsinitial point.

In the case where the group G is finite, g = 0, the unique connection on P is thenull connection and the notion of horizontal lift which we have just defined coincides withthe notion of horizontal lift in a covering.

Given a principal bundle π : P → M , a point m ∈ M and a point p ∈ π−1(m), eachconnection on P determines a multiplicative function from Lm(M) to G, which to each

2.3. YANG-MILLS ACTION AND MEASURE 57

loop l associates holp(ω, l). Changing the reference point p conjugates this application byan element of G.

The action of the gauge group also modifies the mapping l 7→ holp(ω, l) by conjugatingit by an element of G which does not depend on l. Finally, there exists a mapping fromthe quotient A(P )/J (P ) into the quotient M(Lm(M), G)/G.

Proposition 2.2.4 Let π : P →M be a principal G-bundle over M . The mapping

A(P )/J (P ) −→M(Lm(M), G)/G

induced by holonomy is injective.

A random connection should thus determine a random holonomy field. This is preciselythe point of view which has initially motivated the study of processes like those which westudy here. Indeed, physical theories provide us, at a heuristic level, with a probabilitymeasure on the space of connections on a principal bundle, which we are now going todescribe.

2.3 Yang-Mills action and measure

Let π : P → M be a principal G-bundle over M . Let ω be a connection on P . Oneassociates to ω a differential 2-form on P , denoted by Ω and defined by Ω(X, Y ) =dω(X, Y ) + [ω(X), ω(Y )], where the bracket is the Lie bracket of g. One proves that this2-form vanishes as soon as one of its arguments is vertical. It suffices thus to understandit on horizontal vectors. One shows that Ω(X, Y ) = ω([X, Y ]H), where [X, Y ]H is thehorizontal component of the vector [X, Y ]. So, the curvature of ω is identically zero ifand only if the bracket of any two horizontal fields is still horizontal, that is, according toa theorem of Frobenius, if the horizontal distribution is integrable. In the general case,the curvature is a sort of infinitesimal holonomy along small loops. Let us assume thatX and Y are the horizontal lifts of two fields XM and YM on M such that [XM , YM ] = 0.Choose m ∈ M and p ∈ π−1(m). If lε is a small square based at m with two oppositesides following the flow of XM during a time ε and the two other opposite sides followingthe flow of YM during the time ε, then the holonomy of ω along lε computed at p is equalto exp(ε2Ωp(X, Y ) + o(ε2)).

We are now going to define the action of a connection ω as a sort of squared L2 normof its curvature. The surface M is endowed with a surface measure denoted by vol. Forthe sake of simplicity, let us assume that M is orientable so that vol can be identifiedwith a volume 2-form on M . Let then ω be a connection. The curvature of ω, like any2-form on P which vanishes as soon as one of its arguments is vertical, can be written asΩ = Kπ∗vol, where K : P → g is a function. The curvature of a connection inherits fromit a property of equivariance which guarantees us that for all p ∈ P and g ∈ G, we haveK(pg) = ad(g−1)K(p) = g−1K(p)g.

The Lie algebra of G is endowed with a scalar product invariant by conjugation, sothe last equality implies that ‖K(p)‖ = ‖K(pg)‖. The function ‖K‖ is thus constant on


the fibres of P , it identifies with a function on M . We can then define the Yang-Millsaction by setting

S(ω) =

∫M

‖K‖2 dvol.

This definition coincides with the more classical definition S(ω) =∫M〈Ω ∧ ∗Ω〉, in the

case of dimension 2. A fundamental property of S is that it is invariant under the actionof the gauge group : any two connections which differ by a gauge transformation have thesame Yang-Mills action. In particular, this action induces a functional on the quotientspace

S : A(P )/J (P )→ [0,+∞).

The Yang-Mills measure, defined by physicists, is the following measure, defined onthe space of all connections on a given principal bundle.

Definition 2.3.1 (Heuristics) Let M be a surface endowed with a measure of area vol.Let P be a principal G-bundle over M . The Yang-Mills measure on (M, vol, P ) is theprobability measure YMM,vol,P on A(P ) characterised by the expression

YMM,vol,P (dω) =1

ZM,vol,P

e−12S(ω) Dω. (2.1)

The space A(P ) of all connections on a given principal bundle P is an infinite- di-mensional affine space and Dω should be a translation invariant measure on this space.Since the space is not locally compact, such a measure can only be very pathological (itcould for instance be the counting measure). The constant ZM,vol,P is a normalisationconstant, which in this formulation should be infinite because of the action of the gaugegroup, which is not locally compact either and preserves the action S.

One could hope to solve these problems by working on the quotient A(P )/J (P ), ofwhich one could hope that it is small enough, but this space is also non-locally compact.There is no direct way to give a meaning to the definition above.

Nevertheless, if one thinks of the Yang-Mills action as the square of the L2 norm ofthe curvature, the expression (2.1) is analogous to the heuristic expression of a Gaussianmeasure in an infinite-dimensional space. So, one could interpret this expression by sayingthat the Yang-Mills measure is the measure under which the curvature of a connection hasthe distribution of a white noise on (M, vol) with values in the Lie algebra g. If one thenremembers that this curvature is an infinitesimal version of the holonomy along smallloops, it seems natural to take the holonomy field associated by Theorem 1.3.7 to theBrownian motion on G as a candidate for a rigorous definition of the Yang-Mills measure.

2.4 A large deviations principle

2.4.1 Sobolev spaces of connections

In order to give a rigorous meaning to the discussion of the preceding section, we are goingto give a result which relates the Markovian holonomy field associated with the Brownianmotion (which we call the Brownian holonomy field) and the Yang-Mills action. This

2.4. A LARGE DEVIATIONS PRINCIPLE 59

statement takes the form of a large deviation principle. Before entering into the details ofthe formulation of this principle, which still requires some preliminary explanations, werecall briefly the theorem which is the prototype of the result that we will state.

INSERT 8 – Schilder’s theorem [DZ93]

Let C00 ([0, 1]) be the Banach space of continuous real-valued functions on [0, 1], which vanish at 0, endowed

with the uniform norm. The Wiener measure on this space can heuristically be described by the formula

W(df) =1

Zexp

[−1

2

∫ 1

0

f(t)2 dt

]Df.

In this formula a space of functions appears, which is the most natural space of functions for which∫ 1

0f(t)2 dt <

+∞. This space is the Sobolev space W 1,20 ([0, 1]) = H1

0 ([0, 1]) of L2 functions on [0, 1] whose deriva-tive in the distributional sense is a L2 function and which are equal to 0 at 0. Let us define a functionalS : C0

0 ([0, 1]) → [0,+∞] by setting S(f) = ‖f‖2L2([0,1]) if f belongs to H1 and S(f) = +∞ otherwise.Schilder’s theorem is then the following.

Theorem 2.4.1 (Schilder) Let A ⊂ C00 ([0, 1]) be a Borel subset. For all λ ∈ R, set λA = λf : f ∈ A.

Then

− infA

1

2S ≤ lim

T→0log W

(1√TA

)≤ limT→0

log W

(1√TA

)≤ − inf

A

1

2S.

This theorem expresses the fact that the family of distributions of(

(√TBt)t∈[0,1]

)T>0

, where B denotes

the canonical process under W , satisfy a large deviation principle with speed T and rate function 12S as T

tends to 0.Let us recall the four main steps of the proof of this theorem. One first takes care of finite-dimensional

marginals by applying fundamental results of large deviations like Cramer’s theorem. One then gathers thefinite-dimensional results into one single large deviation principle on the space C0

0 ([0, 1]) endowed with thetopology of pointwise convergence, thanks to the Dawson-Gartner theorem. At the end of this step, the ratefunction has the form

S(f) = sup0<t1<...<tn<1

n−1∑i=0

(f(ti+1)− f(ti))2

2(ti+1 − ti).

The third step consists in identifying the rate function with that which we have defined above. One checksimmediately that S(f) ≤ S(f). It suffices thus to prove that S(f) < +∞ implies f ∈ H1

0 and S(f) ≤ S(f). Inorder to prove this point, one considers a sequence (gn)n≥0 of piecewise affine interpolations of f correspondingto a sequence of subdivisions of [0, 1] whose mesh tends to 0. Then one knows explicitly the square H1 normof each gn, it is smaller than 2S(f) which by assumption is finite. The sequence (gn)n≥0 is thus bounded inH1 and one can extract from it a weakly convergent subsequence, to some function h ∈ H1, whose squareof the H1 norm is smaller than 2S(f). Since the injection H1 → C0 is compact, this subsequence convergesuniformly to h, which must then be f . Thus, f belongs to H1 and S(f) ≤ S(f). This finishes the proof ofthe fact that S = S. Finally, the fourth and last step consists in going from the product topology on C0 tothe uniform topology.

We will comment on the step analogous to the third step above in the proof of the large deviation principlefor the Brownian holonomy field. Let us simply emphasise that this step involves two main ingredients : theconstruction (in this case very simple) of piecewise affine approximations of f and a property of compactness.

In order to state a large deviation principle, we must define a rate function and forthis we must identify the space of connections which plays the role of the space H1 in


Schilder’s theorem. This space is, as we have said, the space of connections with finiteenergy. We need to identify a natural space of connections whose Yang-Mills action isfinite. For this, we need to understand more concretely the local nature of this action.

In order to work locally in a principal bundle, one chooses an open subset U of M ,small enough that there exists a local section of P above U , that is, a differentiablemapping σ : U → P such that πσ is the identity of U . One can then consider the 1-formA = σ∗ω on U and the 2-form F = σ∗Ω, both with values in g. The form A writes, inlocal coordinates (x, y), A = Axdx + Aydy, where Ax and Ay are mappings from U to g.The function [Ax, Ay], where the bracket is the Lie bracket of g, that is, the commutatorof matrices, is still a mapping from U to g and the form F writes

F = (∂xAy − ∂yAx + [Ax, Ay])dx ∧ dy.

The contribution of the open subset U to the action S(ω) is thus∫U

‖∂xAy − ∂yAx + [Ax, Ay]‖2 vol(dxdy).

Recall that vol has a smooth positive density with respect to the Lebesgue measure in anychart. The presence of the quadratic term [Ax, Ay] in this expression has the consequencethat the space of connections with finite action is not defined in such a neat way as inthe case of the Brownian motion. Indeed, the expression [Ax, Ay] is meaningless if Ax andAy are distributions, and there is no canonical space of distributional connections whosecurvature is L2. Nevertheless, the action of a connection is finite as soon as its componentshave L2 derivatives and are themselves in L4. Sobolev embedding theorems guarantee usthat the space H1(M) of L2 functions on M with L2 derivatives is contained in Lp(M) forall p ∈ [1,+∞). Incidentally, if M were 3-dimensional (resp. 4- dimensional), we wouldhave H1(M) ⊂ L6(M) (resp. L4(M)), which would be sufficient for the action of a H1

connection to be finite.We will denote by H1A(P ) the space of H1 connections on P . The natural space

of gauge transformations which acts on this space is H2J (P ), the space of H2 gaugetransformations. We claim that the Yang-Mills actions is well defined and finite on thequotient

S : H1A(P )/H2J (P )→ R+. (2.2)

However, in contrast with the case of Schilder’s theorem, the functional which we have justdefined cannot yet be the rate function of our large deviation principle, for it is not definedon the right space. The Brownian holonomy field is not a random connection, but rathera random holonomy. We need thus to define S on the space of multiplicative functions ofloops and for this, we need to make sure that an H1 connection has a holonomy.

2.4.2 The rate functions

The problem is that an H1 connection in two dimensions is far less regular than anH1 function on an interval. In particular, an H1 function on [0, 1]2 is not necessarilycontinuous and cannot be evaluated at a particular point. In order to compute the


holonomy along a loop l : [0, 1]→M of a connection locally given by the form A, we needto be able to solve the differential equation

ht = −htA(lt), h0 = 1,

where h : [0, 1]→ G is the unknown function. The holonomy is then the value of h at 1.

It suffices thus that the function t 7→ A(lt) be L1, as one can check for instance by writingthe solution as a convergent series of multiple integrals. Fortunately, along a curve whichis a smooth submanifold of M , an H1 function admits a trace which is in the space H

12

of this curve, hence in particular in L2, and in L1. This remark would still be true in 3dimensions, where the trace is in L2, but not in 4 dimensions, where the trace of a H1

connection is only in the space H−12 .

Let us denote by P∞(M) the space of paths on M which are concatenations of smoothsubmanifolds. The following result summarises the discussion above and specifies in whichsense an H1 connection is characterised by its holonomy.

Proposition 2.4.2 Let π : P → M be a principal bundle. The mapping induced by theholonomy

H1A(P )/H2J (P ) →M(P∞(M), G)/GM

is injective.

Thanks to this result and (2.2), we can now define a functional on the space of mul-tiplicative functions of loops. In order to avoid working with equivalence classes, weintroduce the following notation.

Definition 2.4.3 Let ω ∈ H1A(P ) be a connection on P . Let h ∈ M(P∞(M), G) be amultiplicative function. We say that ω and h are related, and we write ω ∼ h if the imageof the class of ω by the injection determined by the holonomy is the class of h.

The rate function of the large deviation principle is then the following.

Definition 2.4.4 Let M be a surface without boundary endowed with a measure of area.Let P be a principal G-bundle over M . We define a function IP : M(P∞(M), G) →[0,+∞] by setting

∀h ∈M(P∞(M), G), IP (h) =

12S(ω) if there exists ω ∈ H1A(P ) such that ω ∼ f,+∞ otherwise.

In the case where the surface has a boundary, we want to be able to take boundaryconditions into account. Suppose M given and a set C of boundary conditions in thesense of Definition 1.3.2. We denote by H1AC(P ) the space of H1 connections whichsatisfy these boundary conditions.

Definition 2.4.5 Let M be a surface whose boundary is not empty, endowed with a mea-sure of are and a set C of boundary conditions. Let P be a principal G-bundle over M .We define a function IP,C :M(P∞(M), G)→ [0,+∞] by setting

∀h ∈M(P∞(M), G), IP,C(h) =

12S(ω) if there exists ω ∈ H1AC(P ) such that ω ∼ f,+∞ otherwise.


It is clear that the functions IP and IP,C depend on the bundle P only up to isomor-phism. We can thus index these functionals by isomorphism classes of principalG-bundles.Let us explain how these bundles are classified. Since we are working with the Marko-vian field associated with the Brownian motion, and since the Brownian motion on a Liegroup visits only the connected component of the unit element, we will assume that G isconnected.

INSERT 9 – Classification of principal G-bundles

Let M be a surface and π : P →M a principal G-bundle with G connected. Let us first prove quickly that Pis trivial, that is to say, isomorphic to M ×G, if and only if there exists a smooth mapping σ : M → P suchthat π σ is the identity of M . Such a mapping is called a global section of P . If P is isomorphic to M ×G,let ϕ : P → M × G be an isomorphism. Then σ(m) = ϕ−1((m, 1)) is a global section of P . Conversely,let σ : M → P be a global section of P . For all p ∈ P , the equality π(σ(π(p))) = π(p) implies that thereexists a unique element of G, which we denote by γ(p), such that p = σ(π(p)) · γ(p). Then the mappingϕ(p) = (π(p), γ(p)) is an isomorphism from P onto M ×G.

Let us assume first that the boundary of M is not empty. Then the surface M contains a graph, or abunch of circles, on which it retracts by deformation. For purely topological reasons, any section of P abovethis graph can be extended to a global section of P . On the other hand, since G is connected, the restrictionof P over the graph admits a section. Finally, P is necessarily trivial.

Let us now assume that M is orientable and closed, that is, without boundary. In this case, non-trivialbundles over M may exist. Let us consider a curve C on M which splits M into two surfaces M1 and M2,for example the boundary of a small disk, or a more complicated separating curve. The restrictions of P toM1 and M2 admit global sections, because M1 and M2 are surfaces with boundary. Let us choose two suchsections, one on each side of C. The discrepancy between these two sections can be measured along C, withan appropriate convention on orientation, and determines a closed curve in G. By altering the sections whichwe have chosen in a small cylindrical neighbourhood of C, we can transform this curve in G into any othercurve homotopic to it. It follows easily that that if the construction which we have done with P produces withanother bundle P ′ a curve in G homotopic to that obtained with P , then P and P ′ are isomorphic. Thus,there are at most as many equivalence classes of principal G-bundles as there are homotopy classes in G.

It is slightly less easy to see, but still true, that by changing the sections which we have chosen, we cannotchange the homotopy class of the discrepancy between them. In order to justify this, one can say that anytwo sections over M1 (resp. M2) differ by a smooth mapping from M1 (resp. M2) to G. The point isthat any such mapping sends the boundary of M1 (resp. M2) to a homotopically trivial curve in G. Indeed,the fundamental group of G is Abelian, as that of any topological group. Thus, a mapping from M1 to Gbeing given, the homotopy class of the image of C in G depends only on the image of the homotopy classof C in the quotient π1(M1)/[π1(M1), π1(M1)], which is isomorphic to H1(M1,Z). On the other hand, C isthe boundary of M1 and represents the null homology class (its homotopy class can in fact easily be writtenexplicitly as a product of commutators).

Finally, the homotopy class in G which we obtain by comparing two sections of the restrictions of P onboth sides of a separating curve of M does not depend on the choice of these sections (nor of this curve,by a similar argument) and characterises P up to isomorphism. There are thus exactly as many isomorphismclasses of principal G-bundles over M as there are elements in π1(G). If G is semi-simple, which is equivalentto saying that its centre is trivial, then this fundamental group is finite. Of course, if G is simply connected,then all G-bundles are trivial.

Let us also indicate, although we will not use this fact, that on a closed non-orientable surface, principalG-bundles are classifie by the quotient of π1(G) by its subgroup formed by elements which are squares.

Let us finish this description by giving some examples of fundamental groups of connected Lie groups. For

all N ≥ 1, the fundamental group of U(N) is isomorphic to Z, the homotopy class of a loop t 7→ γt being

characterised by the index of the loop t 7→ det γt with respect to 0 in C. On an orientable surface endowed

with a Riemannian metric, the unitary tangent bundle is a U(1)-principal bundle whose isomorphism class is


given by the Euler characteristic of the surface. For all N ≥ 2, the group SU(N) is simply connected. All

SU(N)- bundles over surfaces are thus trivial3. On the other hand, the centre of SU(N) is formed by scalar

matrices corresponding to N -th roots of unity and the fundamental group of the quotient SU(N)/Z(SU(N)) is

isomorphic to Z/NZ. Since the fundamental group of U(1)d×SU(n1)/Z(SU(n1))× . . .×SU(nr)/Z(SU(nr))

is isomorphic to Zd×Z/n1Z× . . .×Z/nrZ, we see that every finitely generated Abelian group is isomorphic to

the fundamental group of a connected Lie group. Finally, for N ≥ 3, the fundamental group of SO(N) is Z/2Z.

There are thus exactly two non-isomorphic SO(N)-principal bundles over a closed connected orientable surface.

Since all bundles over a surface with boundary are trivial (let us emphasise that thisis no true for non-trivial finite groups, because we have assumed that the group wasconnected), the functional IP,C does not depend on P . We will simply denote it by IC . Inthe case of an orientable surface without boundary, there is thus a functional IP for eachelement of π1(G). We denote by Iz the functional associated with the element z ∈ π1(G).

One can moreover compute the homotopy class of G associated to a bundle P in termsof the holonomy determined by any connection on P . This means that the subsets ofM(P∞(M), G) on which the functionals Iz and Iz′ are finite are the same if z = z′ anddisjoint otherwise.

2.4.3 The Brownian holonomy field and its disintegration

In trying to define a rate function, we have been led to defining many of them, corre-sponding to the various non-isomorphic principal G-bundles over a surface. On the otherhand, the definition of the Brownian holonomy field, just as that of any other Markovianholonomy field, is independent of the choice of a particular G-bundle over M . It is thusnot likely that the Brownian holonomy field such as we have defined it is going to be theobject for which we will be able to state a large deviation principle.

It is in fact possible, and at this point necessary, to disintegrate the Brownian fieldaccording to the isomorphism classes of principal bundles. Before explaining this point,let us briefly describe the Brownian motion on G, in order to correctly define the fieldassociated to it.

INSERT 10 – Brownian motion on a compact connected Lie group

Let G be a connected compact Lie group. Let g be its Lie algebra. Each element of G can be identifiedwith a first-order differential operator on G: for all A ∈ g, one defines the operator LA by setting, for alldifferentiable function f : G→ R and all g ∈ G,

(LAf)(g) =d

dt |t=0f(getA).

Let us now assume that g is endowed with a scalar product invariant by conjugation. Consider an orthonormalbasis (A1, . . . , Ad) of g. Then we define the Laplace operator on G, which is a second-order differential

3This is also true over 3-manifolds. Indeed, H2(G,Z) = 0 for all connected Lie group, so that, by a classicalresult of Hurwitz, a simply connected Lie group has also a trivial second homotopy group: π2(G) = 0.


operator, by

∆ =

d∑i=1

(LA)2.

The Brownian motion on G is the Markov process whose generator is the half of this Laplace operator, whichis just the Laplace-Beltrami operator associated with the bi- invariant metric on G determined by the invariantscalar product on g.

Let us give another description of this Brownian motion as the solution of a stochastic differential equation.We will assume that G is a subgroup of the unitary group U(N) and g, accordingly, is a linear subspace ofu(N), the space of skew-Hermitian matrices.

Let again (A1, . . . , Ad) be an orthonormal basis of g. Let (Bt)t≥0 be the Brownian motion in g, that is,the process whose components on A1, . . . , Ad are independent standard real Brownian motions. Let us defineC = A2

1 + . . .+ A2d. This matrix, called the Casimir matrix, commutes to all elements of G. For example, if

G is U(N) itself and if u(N) is endowed with the scalar product 〈A,B〉 = Tr(A∗B), then C = −NIN . Weconsider the following stochastic differential equation in MN (C), written in Ito notation :

dXt = Xt dBt +1

2CXt dt, X0 = IN .

One checks that the solution (Xt)t≥0 of this equation stays in G and it is called the Brownian motion in G.Let us emphasise that this Brownian motion is not the exponential of the Brownian motion B in g. It

is a non-commutative exponential, also called P -exponential (for path-ordered). One can also say that X isobtained by rolling G on g along B. Finally, for all t ≥ 0, one has

Bt = limn→∞

expX tn

exp(X 2tn−X 2t

n) . . . exp(X (n−1)t

n−Xnt

n).

Let G be a connected compact Lie group endowed with a bi-invariant metric. Let YMdenote the holonomy field associated with the Brownian motion on G.

Theorem 2.4.6 Let (M, vol) be a surface without boundary. There exists a random vari-able

o :M(P(M), G)→ π1(G),

defined under the measure YMM,vol, which measures in a natural sense the isomorphismclass of the fibre bundle underlying a given multiplicative function.

The disintegration of the measure YMM,vol with respect to o determines, for eachz ∈ π1(G), a finite measure YMz

M,vol on M(P(M), G), in such a way that1. YMM,vol =

∑z∈π1(G) YM

zM,vol,

2. for all z ∈ π1(G), on a YMzM,vol(o 6= z) = 0.

It is not difficult to describe the masses of the finite measures YMzM,vol, at least in

the case where M is the sphere S2. In this case, the mass of YMM,vol is Qvol(1), that is,the density at the unit element of G of the distribution at time vol(M) of the Brownianmotion on G issued from the unit element. It is thus the natural mass of the Brownianbridge of length vol(M) on G. If we lift this Brownian bridge to a universal covering of

G, which we denote by G, we find a Brownian motion issued from the unit element and


conditioned to finish at time vol(M) in a point of the fibre over the unit element of G.

This fibre identifies canonically with π1(G) and if we denote by Qvol(M) the density at

time vol(M) of the Brownian motion on G, we can say that the mass of YMzM,vol is equal

to Qvol(M)(z).The construction of the measures YMz

M,vol is the object of the paper [?]. We use aprocedure of discrete approximation analogous to that use in order to define Markovianholonomy fields. However, it is necessary to modify the definition of the discrete theoryin order to take into account the fact that the restriction of a principal bundle to a graphon a surface is always trivial, so that this restriction ignores the topology of P . Onechecks at the end of this constructions that there is a slightly stronger property of mutualsingularity.

Proposition 2.4.7 For all T, T ′ > 0 and all z, z′ ∈ π1(G), the measures YMzM,Tvol and

YMz′

M,T ′vol are mutually singular, unless (T, z) = (T ′, z′).

The discussion of the universal covering of G above is very similar to the discussionin Section 1.3.3. Indeed, in the case without boundary, the random variable T whichappeared there was indeed the same variable which we have called o. Thus, the dis-integration given by Theorem 2.4.6 can be explicitly obtained, in the Abelian case, byreplacing the variable T by the deterministic value corresponding to the topological typeof the chosen bundle. In this Abelian case, where W , the white noise, played the role ofthe curvature, one can in fact understand T as the total curvature of the bundle underconsideration, which, according to Gauss-Bonnet’s theorem, is a topological invariant ofthe bundle.

2.4.4 The large deviation principle

We can now state a rigorous result which links the measures YMzM,vol and the Yang-Mills

action. The following two theorems are proved in [LN5].

Theorem 2.4.8 ([LN5, Theorem 4]) Let (M, vol) be a closed surface endowed with ameasure of area. Let G be a connected compact Lie group endowed with a bi-invariantRiemannian metric. Let z be an element of π1(G). Then the family of probability measuresassociated to the finite measures (YMz

M,Tvol)T>0 on the space (M(P∞(M), G),C ) satisfiesa large deviation principle with speed T and rate function Iz.

Theorem 2.4.9 ([LN5, Theorem 3]) Let (M, vol) be a surface whose boundary is non-empty, endowed with a measure of area and a set C of boundary conditions. Let G bea connected compact Lie group endowed with a bi-invariant Riemannian metric. Thenthe family of probability measures associated to the finite measures (YMM,Tvol,∅,C)T>0 onthe space (M(P∞(M), G),C ) satisfies a large deviation principle with speed T and ratefunction IC.

The first theorem means, by definition of a large deviation principle, that for allmeasurable subset A ⊂M(P∞(M,G)), one has the inequalities

− inff∈A

Iz(f) ≤ limT→0

T logYMzM,Tvol(A) ≤ lim

T→0T logYMz

M,Tvol(A) ≤ − inff∈A

Iz(f), (2.3)


where A and A denote respectively the interior and closure of A for the topology onM(P∞(M,G) which is the trace of the product topology on GP∞(M). Besides, saying thatIz is a rate function implies that it is lower semi-continuous for the same topology. Thistopology is compact. The level sets of Iz are thus automatically compact : it is a goodrate function, in the technical sense of large deviations. However, it is so by the coarsenessof the ambient topology and this precision brings little extra information.

Note that we have stated (2.3) with finite measure rather than probability measures,but the effect of normalisation is harmless at the scale in which we are working, sinceYMz

M,Tvol(1) = O( 1√T

).

The proof of this theorem follows the main steps of the proof of Schilder’s theorem(see Insert 8 above), except that the fourth step is useless, since our results use the prod-uct topology on the space of multiplicative functions. The first step is achieved by usinga classical result of S. Varadhan [Var67a, Var67b] which determines the large deviationfunctional for the heat kernel on G. The second step uses the Dawson-Gartner theorem[DZ93], with a complication due to the fact that all finite-dimensional marginals of holon-omy fields are not explicitly known. It is thus necessary to proceed to approximationsand to check that they are good enough in the exponential scale in which we work. Thethird step is much more difficult than in Schilder’s theorem. The first expression that oneobtains for the rate function is the following :

I(h) = supG

∑F∈F

dG(1, h(∂F ))2

2vol(F ),

where the supremum is taken over the set of all graphs on M . The fact that I(h) is smallerthan I(h) when h is, up to gauge transformation, the holonomy of an H1 connection,follows from an energy inequality initially due to A. Sengupta [Sen98]. One must then

consider an h such that I(h) < +∞, prove that it is associated to an H1 connection and

that I(h) ≤ S(h). For this, we take graphs on the surface and look for connections withminimal energy which have, along the edges of this graph, the same value as h. Theexistence of such connections is established by the following result.

Let G denote a universal covering of G, endowed with a Riemannian metric which

makes the covering map π : G→ G a local isometry. Recall that the fundamental groupof G identifies with the fibre π−1(1).

Proposition 2.4.10 ([LN5, Proposition 22]) Let M be a closed surface. Let G be agraph on M whose edges are smooth submanifolds of M . Let g be an element ofM(E, G).

Let g be an element of M(E, G) such that for all e ∈ E one has π(g(e)) = g(e). Let z bean element of π1(G). Let zF = (zF )F∈F be an element of π1(G)F such that

∏F∈F zF = z.

There exists a principal G-bundle P over M and a connection ω on P such that thefollowing properties are satisfied.

1. The isomorphism class of P is represented by z.2. The connection ω is Lipschitz continuous and its restriction to the interior of each

face is smooth.3. The element of M(E, G) determined by ω is equal, up to gauge transformation, to


g.4. The contribution of each face F to the Yang-Mills action of ω is

SF (ω) =dG(g(∂F )zF )2

vol(F ).

The connection ω whose existence is granted by this proposition is thus the analogueof the piecewise affine approximation of a continuous function in the proof of Schilder’stheorem. This connection has the minimal energy allowed by the energy inequality men-tioned above, given the constraints imposed on its holonomy and on the isomorphismclass of the bundle.

By doing this construction for our element h and for a family of graphs which arefiner and finer, we get a family of connection whose action is bounded by I(h). We neednow to use a compactness result in order to extract a convergent subsequence. But whileit is true and easy to check that the H1 energy of a connection dominates its action, inthat there exists a constant K such that, at least locally, one has S(ω) ≤ K‖ω‖H1 , thereversed inequality is not true. The fact that a set of connections has bounded action doesnot imply that it has bounded H1 norm. Fortunately, a difficult theorem of Uhlenbeckguarantees that one can apply an appropriate gauge transformation to each element ofthe set in such a way that the resulting set has bounded H1 norm. This theorem allowsus to conclude that the two expressions of our rate function coincide.

Theorem 2.4.11 ([Uhl82]) Let P be a principal bundle over M . Let (ωn)n≥1 be a se-quence of H1 connections on P , such that the sequence (S(ωn))n≥0 is bounded. Thenthere exists a H1 connection ω such that, after extracting a subsequence from (ωn)n≥1 andletting a gauge transformation act on each of its terms (the gauge transformation can bedifferent for each term of the sequence), one has the weak convergence

ωnH1

n→∞

ω.

Chapter 3

Unitary Brownian motion

One of the directions of study of the Yang-Mills field which look most appealing to meis the so-called large N limit, that is, the asymptotic behaviour of this field taking itsvalues in the unitary group U(N) as N tends to infinity. This point of view relates theYang-Mills field to the huge field of research constituted by the study of random matricesand their limits. In particular, according to the seminal paper of I. Singer [Sin95], thetheory of free probability should be a natural framework for expressing certain asymptoticproperties of the Yang-Mills field.

The first step in the asymptotic study of the Yang-Mills field consists in studying theasymptotics of the Brownian motion on the unitary group. In this chapter, we are goingto present three distinct works related to this Brownian motion, which correspond to threedistinct aspects of the same object. We will start by recalling what this is about and byrecalling some results which were known before our own work.

This chapter is written so that it might be read independently of the preceding twochapters.

3.1 Combinatorial aspects of the unitary Brownian motion [L6]

3.1.1 Definition of the unitary Brownian motion

Let N ≥ 1 be an integer, which in all the sequel represents the size of the matrices whichwe consider. We consider the unitary group U(N) = U ∈ GLN(C) : UU∗ = U∗U = IN.It is a connected compact subgroup of GLN(C). It is also a smooth real submanifoldof dimension N2 of MN(C). Its Lie algebra is the real linear subspace u(N) of MN(C)formed by skew-Hermitian matrices : u(N) = A ∈ MN(C) : A∗ = −A. It is also theset of matrices A such that etA is unitary for all real t. For all U ∈ U(N), the spaceu(N) is left stable by the conjugation by U , that is, by the transformation A 7→ UAU−1.The linear action of U(N) on u(N) is called the adjoint action, or simply the action byconjugation. We endow u(N) with the following scalar product:

∀A,B ∈ u(N), 〈〈A,B〉〉 = NTr(A∗B).

Let us clarify our notation for the trace: we denote by Tr(·) the usual trace and by tr(·)the normalised trace, so that Tr(IN) = N and tr(IN) = 1. The scalar product 〈〈·, ·〉〉 is

69

70 CHAPTER 3. UNITARY BROWNIAN MOTION

invariant under the action of U(N) by conjugation. It is however not the only one, evenup to a multiplicative constant and the reasons why we have chosen this particular onewill appear soon.

On the Euclidean space (u(N), 〈〈·, ·〉〉), there is a natural Brownian motion, which isthe unique centred Gaussian process (KN(t))t≥0 indexed by R+ and with values in u(N),whose covariance is the following :

∀s, t ∈ R+,∀A,B ∈ u(N), E[〈〈A,KN(t)〉〉〈〈B,KN(s)〉〉] = min(s, t)〈〈A,B〉〉.

One can think of KN as a stochastic process with values in a linear subspace of MN(C)or, if one prefers, as a skew-Hermitian matrix of complex Brownian motions. From thelatter point of view, the entries of KN are complex Brownian motions as independentas the condition of skew-Hermiticity allows them to be. More precisely, each diagonalcoefficient has the distribution of iB/

√N , where B is a standard real Brownian motion,

and each off-diagonal entry has the distribution of (B + iB′)/√

2N , where B and B′ aretwo independent standard real Brownian motions. Finally, the family of entries locatedon the diagonal or above form an independent family. One checks that the matrix ofquadratic covariations 〈dKN , dKN〉t is −IN dt.

One defines the Brownian motion on the group U(N) as the solution of the followingstochastic differential equation :

dUN(t) = UN(t)dKN(t)− 1

2UN(t) dt. (3.1)

This equation is written in the Ito sense and its solution is a process with values inMN(C). One can deduce an equation satisfied by U∗N and, given the quadratic covariation〈KN , KN〉, a simple instance of Ito’s formula implies that d(UNU

∗N) = 0, so that for all

unitary initial condition UN(0), the process UN stays in the unitary group. Most of thetime, we will take UN(0) = IN .

We are now going to describe the generator of UN and write the Ito formula under aform which will be useful. For this, observe that each element of u(N) determines a left-invariant differential operator on U(N). So, if A belongs to u(N), we define a differentialoperator LA by setting, for all differentiable function F : U(N)→ R and all U ∈ U(N),

(LAF )(U) =d

dt |t=0F (UetA).

Let us emphasise that in this formula, the function F is evaluated only at points locatedin the unitary group. Let now X1, . . . , XN2 be an orthonormal basis of u(N). We definethe Laplace operator on U(N) as the second-order differential operator

∆ =N2∑k=1

(LXk)2.

One checks that this operator does not depend on the choice of the orthonormal basis.The Ito formula for UN writes as follows. Observe that, by definition of KN , the processes(〈〈Xk, KN〉〉 : k = 1, . . . , N2) are independent standard real Brownian motions.

3.1. COMBINATORIAL ASPECTS OF THE UNITARY BROWNIAN MOTION [L6] 71

Proposition 3.1.1 (Ito formula) Let F : R+ × U(N) → R be a function of class C2.For all t ≥ 0, one has the equality

F (t, UN(t)) = F (0, UN(0)) +N2∑k=1

∫ t

0

(LXkF )(s, UN(s))d〈〈Xk, KN〉〉s

+

∫ t

0

(1

2∆F + ∂tF

)(s, UN(s)) ds.

Using the stochastic differential equation (3.1) or Ito’s formula, one checks that theBrownian motion is invariant by conjugation and inversion. This means that for allV ∈ U(N), the processes UN , V UNV

−1 and U−1N have the same distribution.

3.1.2 Distribution of the eigenvalues and Newton sums

Let U be an element of U(N). It admits N eigenvalues z1, . . . , zN which are complexnumbers of modulus 1 and we associate to U its empirical spectral measure which is thefollowing probability measure on the unit circle U = z ∈ C : |z| = 1:

µU =1

N

N∑i=1

δzi . (3.2)

For all integer N ≥ 1 and all real t ≥ 0, the measure µUN (t) is a random probabilitymeasure on U and the first result which we are going to present describes partially itsdistribution.

Since U is a compact subset of C, a probability measure ν on U is characterised by itsmoments which are the numbers

∫U z

nν(dz), n ∈ Z. These moments are complex numbersof modulus less than 1. If ν is a random measure, it is characterised by the distribution ofthe sequence of its moments, which in turn, since these moments are uniformly bounded,is characterised by all the expectations of the form

E[∫

Uzm1ν(dz) . . .

∫Uzmrν(dz)

], r ≥ 1,m1, . . . ,mr ∈ Z.

We are going to give a formula for these expectations for the measure µUN (t) when allthe integers m1, . . . ,mr have the same sign. Since the Brownian motion has the samedistribution as its inverse, the measure µUN (t) has the same distribution as its imageby complex conjugation and it suffices to consider the case where m1, . . . ,mr are non-negative.

Observe that the moments of the measure µUN (t) can be written in a simple way interms of UN(t). Indeed, for all n ∈ Z, denoting by z1, . . . , zN the eigenvalues of UN(t),one has ∫

UznµUN (t)(dz) =

1

N

N∑i=1

zni = tr(UN(t)n).

We are thus going to consider quantities of the form tr(UN(t)m1) . . . tr(UN(t)mr) wherem1, . . . ,mr are positive integers. Such a quantity, as a function of the eigenvalues of


UN(t), is a symmetric function, namely a product of Newton sums. The classical notationfor such a product is pλ, where λ is the partition of the integer n = m1 + . . . + mr

formed by m1, . . . ,mr sorted in non-increasing order. However, instead of partitions, weare going to use permutations to index these functions. The fact that this indexation isvery redundant will be compensated by other advantages.

In what follows, we will denote permutations as products of disjoint cycles, omittingcycles of length 1. Thus, the element (124) of S4 is the permutation which sends 1 on 2,2 on 4, 3 on 3 and 4 on 1.

Definition 3.1.2 Let n ≥ 1 be an integer. Let σ be an element of Sn. Let m1, . . . ,mr

be the lengths of the cycles of σ. Let M be an element of MN(C). We set

pσ(M) = tr(Mm1) . . . tr(Mmr)

and Pσ(M) = Tr(Mm1) . . .Tr(Mmr) = N rpσ(M).

More generally, let M1, . . . ,Mn be elements of MN(C). We set

pσ(M1, . . . ,Mn) =∏

(i1...is)cycle of σ

tr(Mi1 . . .Mis)

and Pσ(M1, . . . ,Mn) =∏

(i1...is)cycle of σ

Tr(Mi1 . . .Mis) = N rpσ(M1, . . . ,Mn).

Let us emphasise that the last two definitions make sense thanks to the invariance ofthe trace under cyclic permutation. Thus, for example, if n = 4 and σ = (124), one haspσ(M1,M2,M3,M4) = tr(M1M2M4)tr(M3) = tr(M2M4M1)tr(M3).

3.1.3 Combinatorial expression of the moments of the spectral measure

In order to state the main result of this section, we need to introduce a few notions relatedto a discrete geometry of the symmetric group. Let n ≥ 1 be an integer, Sn the group ofall permutations of 1, . . . , n, and Tn ⊂ Sn the set of all transpositions of Sn. The setTn generates Sn and one forms the corresponding Cayley graph by taking the elements ofSn as vertices and by joining two elements σ1 and σ2 of Sn by an edge if σ1σ

−12 belongs

to Tn.For each permutation σ, we call |σ| the distance between σ and the identity, that is,

the smallest length of a decomposition of σ as a product of transpositions. If `(σ) denotesthe number of cycles of σ, including the cycles of length 1, then |σ| = n−`(σ). In order tounderstand this equality, one can for instance observe what happens when one multipliesa permutation σ by a a transposition (kl). If k and l belong to the same cycle of σ, thenthis cycle is split into two cycles and (kl)σ has one more cycle than σ. On the contrary,if k and l belong to two distinct cycles of σ, then these two cycles coalesce and (kl)σ hasone cycle fewer than σ.


(12) (13) (14) (23) (24) (34)

(123) (132) (124) (142) (134) (143) (234) (243)

(12)(34) (13)(24) (14)(23)

(1234) (1243) (1324) (1342) (1423) (1432)

id

|σ|

0

1

2

3

Figure 3.1: The Cayley graph of S4 generated by its transpositions.

This remark implies also the following consequence: when one follows a path (σ0, . . . , σk)in the Cayley graph, the number of cycles varies at each step by 1 or −1: for eachi = 0, . . . , k − 1, one has |σi+1| = |σi| ± 1.

Definition 3.1.3 Let (σ0, . . . , σk) be a path in the Cayley graph of Sn. By the length ofthis path we mean the integer k and by its defect the integer

d = #i ∈ 0, . . . , k − 1 : |σi+1| = |σi|+ 1.For all σ, σ′ in Sn, all k, d ≥ 0, we denote by Πk(σ) the set of paths of length k issuedfrom σ, by Πk(σ → σ′) the set of paths of length k issued from σ and finishing at σ′, andby S(σ, k, d) the number of paths issued from σ, with length k and defect d.

The defect of a path is thus the number of steps which take it further from the identity.If a path of length k and defect d joins σ to σ′, then we have |σ′| = |σ| − (k − 2d). Inparticular, |k − 2d| ≤ n− 1.

The integer S(σ, k, d) is thus 0 as soon as |k − 2d| ≥ n. On the other hand, thisinteger depends only on the conjugacy class of σ, since the Cayley graph is unchanged ifwe rename all vertices by conjugating them by a fixed permutation.

The main result is the following.

Theorem 3.1.4 ([L6, Theorem 3.3]) Let N, n ≥ 1 be two integers. Let UN be theBrownian motion on U(N) associated to the scalar product 〈〈X, Y 〉〉 = NTr(X∗Y ) onu(N). Let M1, . . . ,Mn be elements of MN(C). Let σ be an element of Sn. Then, for allt ≥ 0, we have the following expansion:

E[pσ(M1UN(t), . . . ,MNUN(t))] = e−nt2

∑k,d≥0

(−t)kk!N2d

∑|σ′|=|σ|−(k−2d)

#Πk(σ → σ′)pσ′(M1, . . . ,MN).

In particular, denoting by m1, . . . ,mr the lengths of the cycles of σ, we have

E[tr(UN(t)m1) . . . tr(UN(t)m1)] = e−nt2

∑k,d≥0

(−t)kk!N2d

S(σ, k, d).


For all T ≥ 0, these series expansion converge normally for (N, t) ∈ N∗ × [0, T ].

The sum over the permutation σ′ which occurs in the first expansion can be rewrittenas a sum over all paths issued from σ with length k and defect d, taking for σ′ thefinishing point of the path. This immediately implies the second expansion when allmatrices M1, . . . ,MN are taken equal to the identity matrix.

3.1.4 Enumeration of paths in the symmetric group

Let us make a few comments about the statement of Theorem 3.1.4. First of all, wehave already observed that the numbers S(σ, k, d) vanish when |k − 2d| > n − 1. Thus,for all k ≥ 0, the contribution of order tk is a polynomial in 1

N2 and, for all d ≥ 0, the

contribution of order 1N2d is polynomial in t.

Then, the numbers S(σ, k, d) are extremely difficult to compute in general. As anamusement, one can check, using Figure 3.1, that S((12)(34), 3, 1) = 104. There is noformula for most of these numbers. The S(σ, k, 0), which count paths which go straightto the identity, are a notable exception. This can be understood by observing that alongthese paths, two elements which do not belong to the same cycle will never be reunited:along these paths, cycles do only split. This indicates that it is enough to know S(σ, k, 0)when σ has only one non- trivial cycle and that in this case, the number depends only onthe length of this cycle, not on the number of fixed points of σ outside this cycle. Thus,it suffices to know the numbers S((1 . . . n), k, 0).

Proposition 3.1.5 ([L6, Proposition 6.6]) For all n ≥ 1 and k ≥ 0, one has

S((1 . . . n), k, 0) =

(n

k + 1

)nk−1.

We recover for example the following classical result: the number S((1 . . . n), n− 1, 0)of ways of writing the n-cycle (1 . . . n) as a product of n− 1 transpositions is nn−2.

Fortunately, the coefficients corresponding to d = 0 are exactly those which matterwhen N tends to infinity. Since the series converges uniformly with respect to N , wecan take the limit under the summation sign and recover the following result, which wasproved in another way by P. Biane [Bia97].

Proposition 3.1.6 For all n ≥ 1 and all t ≥ 0, one has

limN→∞

E[tr(UN(t)n)] = e−nt2

n−1∑k=0

(−t)kk!

(n

k + 1

)nk−1.

We will come back in greater detail to this result. For the time being, let us mentiona formula which expresses the numbers S((1 . . . n), k, d) for all d ≥ 0. Let us recall thedefinition of the Stirling numbers of the second kind, s(n, k), also denoted by

[nk

]. They

are defined by the following identity in C[x]:

x(x− 1) . . . (x− n+ 1) =n∑k=0

[n

k

]xk.


One checks that∣∣[nk

]∣∣ is the number of permutations of Sn at distance n − k from the

identity. In particular,[n0

]= 0. Let us take the convention

[nk

]= 0 if k < 0. We have the

following result.

Proposition 3.1.7 For all n, k, d ≥ 0,

S((1 . . . n), k, d) =1

n

∑r,s,l,m≥0r+s=n−1

l+m=n−1−k+2d

(−1)l+r

r!s!

(n2

(s− r))k [ s+ 1

s+ 1− l

][r + 1

r + 1−m

].

The sum above is a finite sum. It is not obvious that the expression given here whend = 0 coincides with the previous expression - actually I don’t know how to prove itdirectly. Moreover, when d > 0, this formula does not allow one to compute S(σ, k, d)for more general permutations. Indeed, when one counts paths with non-zero defect, onecannot work cycle by cycle as for paths with defect zero.

Nevertheless, this formula allows one to prove the following amusing combinatorialresult.

Proposition 3.1.8 Let n ≥ 1 be an integer. For all integer p ≥ 0, let cn,p be the numberof ways of writing the cycle (1 . . . n) ∈ Sn as a product of p transpositions. The numbercn,p is non-zero only if p = n− 1 + 2d for a certain d ≥ 0. In this case,

cn,p = S((1 . . . n), n− 1 + 2d, d) =np

n!

n−1∑r=0

(−1)r(n− 1

r

)(n− 1

2− r)p

.

For all n ≥ 1, one has the equality∑p≥0

cn,pxp

p!=

1

n!en(n−1)

2x(1− e−nx

)n−1.

In particular, cn,n−1 = nn−2, cn,n+1 = 124

(n2 − 1)nn+1 and

cn,n+3 =1

5760(5n− 7)(n+ 3)(n+ 2)(n2 − 1)nn+3.

3.1.5 Unitary Brownian motion and symmetric random walk

In this paragraph, we are going to explain how Schur-Weyl duality makes Theorem 3.1.4plausible, and state a result which relates the Brownian motion on U(N) and the mostnatural random walk on the symmetric Sn.

INSERT 11 – Schur-Weyl duality

Let n,N ≥ 1 be two integers. On the vector space (CN )⊗n of n- linear forms on (CN )∗, there are twonatural actions ρ and π, respectively of the unitary group U(N) and the symmetric group Sn, given by

ρ(U)(x1 ⊗ . . .⊗ xN ) = Ux1 ⊗ . . .⊗ Uxnπ(σ)(x1 ⊗ . . .⊗ xN ) = xσ−1(1) ⊗ . . .⊗ xσ−1(n).


These actions commute to each other: for all U ∈ U(N) and all σ ∈ Sn, one has ρ(U) π(σ) = π(σ) ρ(U).Let R denote the subalgebra of End((CN )⊗n) generated by the (ρ(U) : U ∈ U(N)) and P that generated bythe (π(σ) : σ ∈ Sn). Then each element of R commutes to each element of P. Schur-Weyl’s theorem statesthat the algebras R and P are each other’s commutant.

Theorem 3.1.9 (Schur-Weyl duality) Each endomorphism of (CN )⊗n which commutes to R belongs to Pand each endomorphism of (CN )⊗n which commutes to P belongs to R.

By the Jacobson density theorem, which implies that a subalgebra of the algebra of endomorphisms of afinite-dimensional vector space is equal to the commutant of its commutant, the two assertions of the theoremare equivalent.

The algebra P is simply the algebra of the linear combinations of the π(σ) where σ runs over Sn. Inorder to describe it, it is convenient to introduce the group algebra C[Sn] of Sn, which is the vector space offormal linear combinations

∑σ λσσ of elements of σ endowed with the multiplication∑

σ

aσσ∑σ′

bσ′σ′ =

∑σ,σ′

aσbσ′σσ′.

This algebra identifies naturally with the convolution algebra of functions on Sn. The action π : Sn →End((CN )⊗n) extends by linearity to C[Sn] to give a morphism of algebras whose range is precisely P.

We are going to apply Schur-Weyl duality to the image by ρ of the Laplace operatoron U(N). In order to give a meaning to this image, we start by extending ρ to u(N),by differentiation: for all A ∈ u(N), one can define ρ(A) = d

dt |t=0ρ(etA) = (LAρ)(IN).

Concretely, one has

ρ(A)(x1 ⊗ . . .⊗ xN) =N∑k=1

x1 ⊗ . . .⊗ xk−1 ⊗ Axk ⊗ xk+1 ⊗ . . .⊗ xN .

One can then define

ρ(∆) =N2∑k=1

ρ(Xk)2,

which is none other than (∆ρ)(IN) if we see ρ as a smooth mapping from U(N) into thevector space End((CN)⊗n).

The operator has the property of being invariant under translations. This can bechecked infinitesimally by checking that it commutes to all operators LA, where A ∈ u(N).This implies that the endomorphism ρ(∆) of (CN)⊗n commutes toR. According to Schur-Weyl’s theorem, it belongs to P and can thus be written as a linear combination of theπ(σ) for σ ∈ Sn.

Let us call L the generator of the Markov chain on the Cayley graph of the symmetricgroup which jumps at rate 1

Nfrom its present state to each of its n(n− 1)/2 neighbours.

The operator L acts on functions from Sn into C according to the formula

Lf(σ) = −n(n− 1)

2Nf(σ) +

1

N

∑τ∈Tn

f(στ).


If we identify the space of functions from Sn into C with the algebra C[Sn], the actionof L is simply the multiplication by the element

L = −n(n− 1)

2N+

1

N

∑τ∈Tn

τ.

The result, implicitly present in the work of D. Gross and W. Taylor [GT93a], is thenthe following.

Lemma 3.1.10 One has the equality

1

2ρ(∆) + π(L) = −n

2− n(n− 1)

2N.

This result is remarkable because it relates two objects which have a probabilisticinterpretation. To the expense of a small algebraic manipulation, it allows us to provethe following result.

Proposition 3.1.11 Let (σt)t≥0 be the Markov chain on Sn with generator L. Then theprocess (

ent2

+n(n−1)t

2N Pσt(UN(t)))t≥0

is a martingale. In particular, for all t ≥ 0, one has the equality

E[Pσt(UN(t))] = e−nt2−n(n−1)t

2N E[N `(σ0)],

where `(σ0) denotes the number of cycles of σ0.

This result, apart from the fact that it relates the Brownian motion on the unitarygroup to a random walk on the symmetric group, is the key to the proof of Theorem 3.1.4.Indeed, by letting the Markov chain (σt)t≥0 start from the various elements of Sn, onegets enough information to find E[Pσ(UN(t))] for all σ. It is then the change from Pσ topσ which forces one to consider the number of cycles of the permutations and to let thedefect of paths into the picture.

3.1.6 Random ramified coverings

In the two papers [GT93a, GT93b], D. Gross and W. Taylor state many formulas whichrelate on one hand partition functions of the Yang-Mills measure on surfaces and withvalues in the unitary group and on the other hand average values of geometric quantitiesassociated to ramified coverings over these surfaces. In some cases, singularities otherthan ramification points are authorised for these coverings. However, in the simplestcase, where the surface is a disk, the partition function is simply the heat kernel onthe unitary group and the coverings considered by Gross and Taylor are good ramifiedcoverings. Theorem 3.1.4 allows one to prove rigorously their formula.

Let D be the unit disk of the plane R2. We need to describe the probability measureon the set of ramified coverings which occur in this formula. It is not exactly of the sameform as those which we have constructed in Section 2.1.2, but it looks very much like it.


The finite group which we consider here is the symmetric group Sn. The first differencewith Section 2.1.2 is that instead of considering ramified Sn-principal bundles, we considerramified coverings of degree n, without action of any group. There is however a bijectivecorrespondence between these two types of coverings, analogous to the correspondencebetween the frame bundle and the tangent bundle of a smooth manifold.

Let π : R→ D be a ramified covering of degree n. Let y ∈ D be a ramification point.Let l be a loop in D which goes around y. Let us denote by x the base point of l. Ifwe label from 1 to n the points of the fibre π−1(x) = x1, . . . , xn, then the mappingσ : 1, . . . , n → 1, . . . , n which to each i associates the label of the endpoint of the liftof l issued from xi is a bijection. Changing the labelling of π−1(x) modifies this bijectionby conjugating it by another bijection. The conjugacy class of σ is in fact independent ofall the choices which we have made, in particular that of l, and characterises the type oframification at y. It is called the monodromy at y. We say that a ramification point isgeneric if the monodromy at this point is the conjugacy class of a transposition. This isequivalent to the equality #π−1(y) = n− 1.

The restriction of π : R → D to the boundary of D is a covering of the unit circle,which is characterised by a conjugacy class of Sn, or equivalently a partition of the integern, but we will stick to permutations.

Definition 3.1.12 Let D be the unit disk of R2. Let n ≥ 1 be an integer. Let σ be apermutation of Sn. Let X be a finite subset of D \ 0. We denote by Rn,X,σ the set oframified coverings π : R→ D of degree n such that the following properties are satisfied.

1. For all x ∈ X, the covering R admits at x a generic ramification, that is, #π−1(x) =n− 1.

2. The covering R is ramified at no point of D \ (X ∪ 0).3. The monodromy of R the boundary of D is the conjugacy class of σ.

We denote by Rn,σ the union of the Rn,X,λ where X runs over the set of finite subsets ofD \ 0.

There is no restriction on the ramification at 0: it can be a regular point or a pointwith an arbitrary ramification. This is what Gross and Taylor call an Ω-point.

For all finite subset X of D\0, the set Rn,X,σ is finite. We endow it with the uniformmeasure

ρn,X,σ =∑

R∈Rn,X,σ

n!

Aut(R)δR.

Let us now consider a measure of area vol on D, with total area equal to 1, for example1π

times the Lebesgue measure. For all t, let Ξt be a Poisson point process on D withintensity tvol, which we see as a probability measure on the set X of finite subsets ofD \ 0. We then set

ρtn,σ =

∫Xρn,X,σ Ξt(dX).

The measure ρtn,σ is a finite measure, with total mass et(n2)−t. We denote by µtn,σ the

corresponding probability measure on Rn,σ.

3.2. ASYMPTOTIC BEHAVIOUR OF THE EIGENVALUES [LM9] 79

Finally, for all π : R → M in Rn,σ, we denote by k(R) the number of ramificationpoints of R other than 0, and by χ(R) the Euler characteristic of R. The formula of Grossand Taylor is now the following.

Theorem 3.1.13 ([L6, Theorem 8.3]) Let N, n ≥ 1 be integers. Let σ be an elementsof Sn. Let t ≥ 0 be a real. Then one has the equality

E[Pσ(UN(t))] = e−nt2

+(n2)t∫Rn,σ

(−1)k(R)Nχ(R) µtn,σ(dR).

This theorem gives a geometric explanation for the fact that only non-negative evenexponents of 1

Nappear in Theorem 3.1.4. Indeed, if we divide both sides of the equality

by N `(σ), where `(σ) is the number of cycles of σ, we see that E[pσ(UN(t))] expands inpowers of N of the form Nχ(R)−`(σ). Observe that the boundary of a covering R ∈ Rn,σ hasexactly `(σ) connected components. So, on one hand, χ(R)− `(σ) has the same parity asχ(R)+`(σ) which is the Euler characteristic of the closed surface obtained by gluing a diskalong each connected component of its boundary, and is thus an even integer. On the otherhand, let c(R) denote the number of connected components of R, which is also the numberof connected components of the closed surface R′ described in the last sentence. Then`(σ) ≥ c(R) = c(R′), because from each regular point of R one can reach the restriction ofR over the boundary of the unit circle, for example by following the horizontal lift of a well-chosen path. So, χ(R)−`(σ) = χ(R)+`(σ)−2`(σ) = χ(R′)−2`(σ) ≤ χ(R′)−2c(R′) ≤ 0.The powers of N which appear are thus non-negative even integers.

When N tends to infinity, only ramified coverings with maximal characteristic con-tribute : these are the coverings by a finite union of disks.

Let us conclude this section by emphasising that the link which this theorem revealsbetween the Yang-Mills field and random ramified coverings is not of the same nature asthe link which we have described in Section 2.1, which was a link of analogy. The linkrevealed by the formula of Gross and Taylor is a link of duality, which can be understoodas a consequence of Schur-Weyl duality.

3.2 Asymptotic behaviour of the eigenvalues [LM9]

3.2.1 Distribution of the eigenvalues : asymptotic results

Proposition 3.1.6 indicates that the intensity of the random measure µUN (t) has, for allt ≥ 0, a limit as N tends to infinity. Let us start by describing this limit. We setU = z ∈ C : |z| = 1.

Proposition 3.2.1 For all t ≥ 0, there exists a unique probability measure νt on U suchthat for all n ≥ 0 one has∫

Uξnνt(dξ) =

∫Uξ−nνt(dξ) = e−

nt2

n−1∑k=0

(−t)kk!

(n

k + 1

)nk−1.


This measure is also characterised by the fact that, for all complex number z in a neigh-bourhood of 0, one has the equality∫

U

1

1− zz+1

etzet2 ξνt(dξ) = 1 + z.

For all t > 0, the measure νt admits a density with respect to the uniform measureon U, which is analytic inside its support. This support is an interval which contains 1,symmetric with respect to the real axis, which grows with t and covers the whole circleU precisely at time t = 4 (see [Bia97]).

It is however impossible to express the density of νt with usual functions only, unlessone includes in such functions the Lambert W function, which is the reciprocal functionof z 7→ zez.

Figure 3.2: Density of νt at eiθ in function of θ ∈ [−π, π] and t. The support of the measure is knownexplicitly at each time. This support is the entire circle U if and only if t ≥ 4. The regularity of thedensity of νt changes when t crosses this value, from Holder continuous with exponent 1

2 for t < 4 to realanalytic for t > 4. This is indeed a phase transition. For t > 4, one sees on the picture that the densityconverges rapidly to 1. This convergence is in fact exponential, uniformly in θ.

A convenient way of expressing limit theorems for the distribution of eigenvalues of uni-tary matrices whose size tends to infinity is to consider a function f : U→ R which is regu-lar enough and to associate to it the sequence of real random variables (tr(f(UN(t))))N≥1.Proposition 3.1.6 implies easily the following result.

Proposition 3.2.2 Let f : U→ R be a continuous function. Then for all t ≥ 0 one hasthe convergence

E[trf(UN(t))] −→N→∞

∫Uf dνt.


In the case where the matrix is picked under the Haar measure, the invariance undertranslation of this measure implies immediately the following result.

Lemma 3.2.3 Let f : U → R be a continuous function. For all N ≥ 1, let VN be arandom unitary matrix distributed according to the Haar measure on U(N). Then for allN ≥ 1, one has

E[trf(VN)] =

∫Uf(ξ) dξ.

This lemma can be seen as a limit of the previous one when t tends to +∞. Let usobserve that, since the average repartition of the eigenvalues of VN is uniform on the circle,the random variable is defined almost surely as soon as f is defined almost everywhere.Thus, the previous lemma is still meaningful, and true, for a function f ∈ L1(U).

P. Diaconis and S. Evans have studied the fluctuations of trf(VN) and they have

established a central limit theorem [DE01]. The Sobolev space H12 , the definition of

which is recalled below, plays an essential role in this theorem. For all integrable functionf on U and all n ∈ Z, we denote by f(n) the n-th Fourier coefficient of f .

Definition 3.2.4 The space H12 (U) is the subspace of L2(U) defined by

H12 (U) = f ∈ L2(U) :

∑n∈Z

|n||f(n)|2 < +∞,

endowed with the Hilbert scalar product

〈f, g〉2H

12

= f(0)g(0) +∑n∈Z∗|n|f(n)g(n).

The elements H12 are not necessarily continuous functions. The result of Diaconis and

Evans is the following.

Theorem 3.2.5 ([DE01]) Let f1, . . . , fn : U→ R be functions of H12 such that

∫U fi(ξ) dξ =

0 for all i ∈ 1, . . . , n. Let Σ be the n × n non-negative symmetric matrix defined byΣ(f1, . . . , fn) = (〈fi, fj〉H 1

2)i,j=1,...,n. Then, with the notation of Lemma 3.2.3, one has, as

N tends to +∞, the following convergence in distribution:

N(tr(f1(VN)), . . . , tr(fn(Vn)))(law)−→

N→+∞N (0,Σ(f1, . . . , fn)).

With Mylene Maıda, we have established an analogous theorem for random variablesof the form trf(UN(t)). The expression of the covariance in this case is however morecomplicated. It requires the introduction of some notions of free probability.

INSERT 12 – Free probability and free unitary Brownian motion [VDN92, ?, Bia97]

Free probability is a non-commutative analogue of the theory of probability, in which the notion of independenceis replaced by the notion of freeness.


Definition 3.2.6 A non-commutative probability space is a pair (A, τ) where A is a complex involutive unitalalgebra and τ is a linear form on A which is

1. positive : τ(a∗a) ≥ 0 for all a ∈ A,2. normalised : τ(1) = 1,3. tracial : τ(ab) = τ(ba) for all a, b ∈ A.

The form τ is also called a state.

The two fundamental examples of non-commutative probability spaces are on one hand (L∞(Ω,P),E),where (Ω,P) is a classical probability space, and on the other (MN (C), tr). Their tensor product is a newnon-commutative probability space, the space of N ×N random matrices with bounded entries endowed withthe state E[tr(·)]. The main definition of free probability theory is the following.

Definition 3.2.7 Let (A, τ) be a non-commutative probability space. Let A1, . . . ,Ar be involutive subal-gebras of A. The family A1, . . . ,Ar is said to be free with respect to τ , or simply free, if the followingproperty is satisfied.

(F) For all integer n ≥ 1, for all sequence of integers i1, . . . , in which belong to 1, . . . , r and such thati1 6= i2, i2 6= i3, . . . , in−1 6= in, for all a1, . . . , an such that a1 ∈ Ai1 , . . . , an ∈ Ain and τ(a1) = . . . =τ(an) = 0, one has

τ(a1 . . . an) = 0.

Elements b1, . . . , br of A are said to form a free family if the involutive subalgebras which they generate forma free family.

The examples of non-commutative probability spaces which we have already given are not rich enough toprovide us with non-trivial examples of free families. For example, two elements of (L∞(Ω,P),E), that is tosay, two classical random variables, are free if and only if one of them is constant P-almost surely. This followsfrom the equalities

τ(ab) = τ(a)τ(b) and τ(abab) = (τ(a2)− τ(a)2)τ(b)2 + τ(a)2(τ(b2)− τ(b)2) + τ(a)2τ(b)2, (3.3)

which are true for any two free elements of a non-commutative probability space. Indeed, if X and Y are freeand centred, the first equality applies to X2 and Y 2 and the second applied to X and Y give two expressionsof E[X2Y 2] whose equality implies that X or Y is almost surely equal to zero.

In the case of (MN (C), tr), one can also show that if two matrices are free, then one of them is scalar.This exercise seems not to be solved in many places in the literature, so we pause briefly to indicate the stepsof a proof. Consider two matrices A and B which are free. Let us start by assuming that A is a non-scalarprojector and B is Hermitian and traceless. The formulas (3.3) imply the relations

tr(AB2) = 0 , tr(ABAB) = tr(A)2tr(B2) and tr((IN −A)B(IN −A)B) = (1− tr(A))2tr(B2)

from which it follows, since 0 < tr(A) < 1, that tr(B2) = 0 and B = 0. It follows immediately that if Ais a non-scalar projector and B is Hermitian or skew-Hermitian, then B is scalar. Since the assumption thatA and B are free implies that A is free with any polynomial in B and B∗, we find that a matrix which isfree with a non-scalar projector is scalar. Finally, if we suppose only that A is not scalar, then one of the twodiagonalisable matrices A+A∗ and A−A∗ admits two distinct eigenvalues, so that there exists a polynomialin A and A∗ which is a non-scalar projector. The matrix B is free with this projector, hence scalar.

The simplest example of two free non-scalar elements is the following, which in fact has motivated thedefinition of the notion of freeness by D. Voiculescu. Let F2 be the free group on the set x, y. The elementsof F2 are thus reduced words in the letters x, y and their inverses, and one multiplies them by concatenatingthem and reducing them. We denote by 1 the empty word, which is the unit element. We consider the algebraA = C[F2] of formal finite linear combinations of elements of F2 with complex coefficients, endowed with theinvolution (

∑w cww)

∗=∑w cww

−1. We endow A with the linear form

τ

(∑w

cww

)= c1.


One checks easily that τ is a state. We have thus constructed a new example of a non-commutative probabilityspace. In this space, the elements x and y are free, by definition of the free group F2.

Let (A, τ) be a non-commutative probability space. We say that an element a ∈ A is self-adjoint ifa = a∗ and an element u is unitary if uu∗ = u∗u = 1. The positivity condition which is in the definition ofa state implies that if a is self-adjoint, then the sequence (τ(an))n≥0 is the sequence of the moments of aprobability measure on R which, if it is unique, is called the distribution of a. A condition which suffices toensure the unicity of this measure is that A is a Banach algebra and τ is continuous. Indeed, in this case,τ(an) = O(‖a‖n), which implies that the distribution of a has compact support. Similarly, if u is unitary,the sequence (τ(un))n∈Z is the sequence of the Fourier coefficients of a probability measure on U, also calledthe distribution of u. One checks successively, in the three examples that we have given, that the non-commutative distribution of a real random variable coincides with its usual distribution, that the distributionof a Hermitian or unitary matrix is its empirical spectral measure (see (3.2)) and that the elements x and yof C[F2], which are unitary, have the uniform measure on U as their distribution.

The notion of distribution in free probability identifies with the notion of moments. By definition, thedistribution of a unitary element u, for example, is the linear form on C[t, t−1] which to a polynomial Passociates τ(P (u, u−1)). In order to describe the joint distribution of several elements, one uses polynomialswith several non-commuting indeterminates. We denote for example by C〈t1, . . . , tn, t−11 , . . . , t−1n 〉 the algebraof polynomials with n indeterminates and their inverses, which can be identified naturally with C[Fn], thegroup algebra of the free group on n letters.

Definition 3.2.8 The joint non-commutative distribution of n unitary elements u1, . . . , un is the linear formon C〈t1, . . . , tn, t−11 , . . . , t−1n 〉 which to a polynomial P associates τ(P (u1, . . . , un, u

−11 , . . . , u−1n ).

The convergence in distribution is also defined as the convergence of all moments.

Definition 3.2.9 Let (A, τ) and (Ar, τr), r ≥ 1 be non- commutative probability spaces. Let (ui)i∈I and(ui,r)i∈I , r ≥ 1 be families of elements of these spaces indexed by the same set I. We say that the sequence offamilies (ui,r)i∈I , r ≥ 1 converges in non-commutative distribution to (ui)i∈I if for all n ≥ 1, all i1, . . . , in ∈ Iand all P ∈ C〈t1, . . . , tn, t−11 , . . . , t−1n 〉 one has the convergence

τr(P (ui1,r, . . . , uin,r, u−1i1,r

, . . . , u−1in,r)) −→r→∞ τ(P (ui1 , . . . , uin , u−1i1, . . . , u−1in )).

We can now define the free unitary Brownian motion, which is a particular process with stationary freeincrements.

Definition 3.2.10 Let (A, τ) be a non-commutative probability space. A free multiplicative Brownian motionis a family (ut)t≥0 of unitary elements of A which satisfies the following properties.

1. For all 0 ≤ t1 ≤ . . . ≤ tn, the elements ut1 , ut2u∗t1 , . . . , utnu

∗tn−1

are free.2. For all 0 ≤ s ≤ t, the distribution of utu

∗s is equal to the distribution of ut−s.

3. For all t ≥ 0, the distribution of ut is the probability measure νt on U.

This free process is the limit in distribution in the sense of Definition 3.2.9 of the unitary Brownian motion.

Proposition 3.2.11 The sequence of families of random variables ((UN (t))t≥0)N≥0, where each UN (t) is

seen as an element of the non-commutative probability space (L∞(Ω,P)⊗MN (C),E⊗ tr) for an appropriate(Ω,P), converges in non-commutative distribution to a free unitary Brownian motion.

It is not obvious, but true, that there exists a free unitary Brownian motion. One can even construct anarbitrary finite number of them in such a way that the algebras which they generate are mutually free.

The covariance of the fluctuations of the random variables of the form trf(UN(t)) isgiven by the following bilinear form.


Definition 3.2.12 Let (A, τ) be a C∗-non-commutative probability space in which thereexists three free multiplicative Brownian motions u, v, w which are mutually free. LetT ≥ 0 be a real. Let f, g : U→ R be two functions of class C1. We define

σT (f, g) =

∫ T

0

τ(f ′(usvT−s)g′(uswT−s)) ds.

The assumption that (A, τ) is a C∗-non-commutative probability space means that Ais a C∗-algebra, which allows us, for all unitary element u ∈ A and all continuous functionf : U→ R, to define, by functional calculus, a new element f(u) of A. The central limittheorem, analogous to the theorem 3.2.5 of Diaconis and Evans, is the following.

Theorem 3.2.13 ([LM9, Theorem 2.6]) Let UN be the Brownian motion on U(N)associated to the scalar product on u(N) given by 〈〈X, Y 〉〉 = NTr(X∗Y ). Let T ≥ 0be real. Let n ≥ 1 be an integer. Let f1, . . . , fn : U → R be functions of class C1(U)whose derivatives are Lipschitz continuous. The n × n real symmetric matrix defined byΣT (f1, . . . , fn) = (σT (fi, fj))i,j∈1,...,n is non-negative. Moreover, as N tends to infinity,the following convergence in distribution of random vectors of Rn holds:

N (trfi(UN(T ))− E [trfi(UN(T ))])i∈1,...,n(law)−→N→∞

N (0,ΣT (f1, . . . , fn)). (3.4)

The regularity of the functions that we consider is not there to ensure the positivity ofthe matrix ΣT , which is true already for C1 functions, but rather to ensure that our proofworks, since it relies on arguments of stochastic calculus. However, the result of Diaconisand Evans and the convergence of the Brownian motion to its invariant measure suggestthat it should be possible to improve this result by enlarging the class of functions forwhich the result holds. We have at least established the following result for the covariance.

Theorem 3.2.14 ([LM9, Theorem 8.3]) For all pair (f, g) of functions of class C1 onU and whose derivatives are Lipschitz continuous, one has, with the notation of Theorems3.2.5 and 3.2.13, the following convergence:

σT (f, g) −→T→∞

〈f, g〉H

12.

We have in fact obtained a better result by showing that, for T large enough, one canextend the definition of σT (f, g) to all pairs of function (f, g) which belong to the space

H12 and that with this extended definition, the previous result still holds. More precisely,

denoting, for all j, k ∈ Z and all T ≥ 0,

τj,k(T ) =

∫ T

0

τ(

(usvT−s)j (uswT−s)

k)ds,

we have been able to prove a bound which guarantees us that, for T large enough, if fand g are two functions in H

12 , then the series

σT (f, g) = −∑j,k∈Z

jkf(j)g(k)τj,k(T )

3.3. BETWEEN INDEPENDENCE AND FREENESS [LBG8] 85

1 2 3 4 5 6

0.2

0.4

0.6

0.8

1.0

1 2 3 4 5 6

0.5

1.0

1.5

2.0

2.5

1 2 3 4 5 6

1

2

3

4

1 2 3 4 5 6

-1.0

-0.5

0.5

1.0

Figure 3.3: For all k ≥ 1, set sk(eiθ) = sin(kθ) and ck(eiθ) = cos(kθ). These pictures represent functionsof T for T ∈ [0, 6]. Top left : µk(T ) for k ∈ 1, . . . , 6. Top right : σT (sk, sk+1) for k ∈ 1, . . . , 15.Bottom left : σT (sk, sk) and σT (ck, ck) for k ∈ 1, . . . , 8. Bottom right : σT (sk, sk+2) for odd k in1, . . . , 13.

converges. Moreover, on one hand, if f and g have Lipschitz continuous derivatives, thenthe two definitions of σT (f, g) which we have given coincide, and on the other hand, theconvergence stated by Theorem 3.2.14 still holds with this new definition of σT .

Let us emphasise that the extended definition of σT is valid only for T large enough(say T ≥ 32). Generally speaking, it is very difficult to control moments or covariances insmall time. This could be compared with the fact that the measure νt, the distribution ofthe free unitary Brownian motion, has its support strictly contained in U for t < 4 and ananalytic density for t > 4. Figure 3.3 illustrates the fairly mysterious behaviour in smalltime of some of the functions which we have considered.

3.3 Between independence and freeness [LBG8]

With Florent Benaych-Georges, we have used the unitary Brownian motion to proposean interpolation between the notions of independence and freeness for elements of a non-commutative probability space. We are going to introduce this question from the pointof view of the convolution of measures.


3.3.1 Convolutions

Let µ and ν be two probability measures on R. The classical convolution of µ and ν,denoted by µ ∗ ν, is another probability measure on R which can be described as thedistribution of the sum of two independent random variables of respective distributionsµ and ν. Another way of describing this measure is the following. Let (AN)N≥1 and(BN)N≥1 be two sequences of diagonal matrices such that for all N ≥ 1, AN and BN havesize N × N and, as N tends to infinity, the empirical spectral measures µAN and µBNconverge respectively to µ and ν. For all N ≥ 1, let SN be a random matrix distributeduniformly on the set of the N ! permutation matrices of size N . Then the followingconvergence in distribution holds:

µAN+SNBNS−1N

(law)−→N→∞

µ ∗ ν.

For all N ≥ 1, let us now consider a matrix VN uniformly distributed on U(N). Then wehave the following convergence:

µAN+VNBNV−1N

(law)−→N→∞

µ ν,

where µ ν is the measure on R which can be described as the distribution of the sumof two free self-adjoint elements of a non-commutative probability space whose respectivedistributions are µ and ν. This measure depends only on µ and ν since, just as forindependence in the classical case, the joint distribution of two free elements is completelydetermined by their individual distributions.

For all N ≥ 1, there is a natural interpolation between the distribution of SN and thatof VN : it is the family of the distributions of UN(t)SN where t runs over R+. For t = 0,we have UN(0)SN = SN and, when t tends to +∞, UN(t)SN converges in distribution toa uniform unitary matrix.

Theorem 3.3.1 ([LBG8, Corollary 2.10]) Let µ and ν be probability measures on R.Let (AN)N≥1 and (BN)N≥1 be two sequences of N × N diagonal matrices as above. LetUN be a Brownian motion on U(N). Let t ≥ 0 be a real. Then, as N tends to infinity, theempirical spectral measure of the matrix AN + UN(t)SNBNS

−1N UN(t)−1 converges weakly

in probability to a deterministic measure on R which depends only on µ, ν and t, whichwe denote by µ ∗t ν and call t-free convolution of µ and ν.

A classical example where one can compute explicitly the free convolution of twomeasures is that where µ = ν = 1

2(δ1 + δ−1). In this case, µ ∗ ν = 1

4δ−2 + 1

2δ0 + 1

4δ2 and

µ ν = 1[−2,2](x) dxπ√

4−x2 , a dilation of the arcsine law. One can show that for all t > 0,

δ1 + δ−1

2∗tδ1 + δ−1

2= 1[−2,2](x)

ρ4t(e4i arccos x

2 )

π√

4− x2dx, (3.5)

where, for all t > 0 and all θ ∈ R, ρt(eiθ) is the density at eiθ of the measure νt with

respect to the uniform measure on U.


Figure 3.4: Density of δ1+δ−1

2 ∗t δ1+δ−1

2 at x ∈ [−2, 2] in function of x and t. One can describe completelythe support of the measure at each time. The first time at which this support is the whole interval [−2, 2]is t = 1 and the last two points to enter this support are −

√2 and

√2.

3.3.2 Dependence structures and t-freeness

Just as in the case of independence and freeness, the existence of the t-free convolution oftwo measures is a by-product of the existence of a structure of dependence between sub-algebras of a non-commutative probability space. By a structure of dependence, we meanthe following: in a non-commutative probability space (A, τ) where two subalgebras A1

and A2 are given, a way of reconstructing the restriction of τ to the subalgebra generatedby A1 ∪ A2 from the restrictions of τ to A1 and A2. In commutative terms, we wouldsay that such a structure allows one to infer the joint distribution of a family of randomvariables from the knowledge of their individual distributions.

INSERT 13 – Dependence structures and universal models

A way of describing a structure of dependence between two subalgebras A1 and A2 of a non-commutativeprobability space is giving a rule which allows one to compute τ(P (a1, a2)) for all a1 ∈ A1, a2 ∈ A2 andall polynomial P ∈ C〈t1, t2〉 given the moments of a1 and a2. It is exactly in this way that we have definedfreeness1 (see Definition 3.2.7). The rule E[f(X)g(Y )] = E[f(X)]E[g(Y )] which characterises independenceof random variables is also of this nature. Such a description does however not ensure that there exist non-trivial examples of the structure of dependence which one is defining. The example of two free elements whichwe have given in Insert 12, albeit natural, was not contained in the definition of freeness.

In classical probability theory, the construction which guarantees the existence of independent variableswith arbitrary distributions is the Cartesian product of measurable spaces endowed with the tensor product

1It would be more honest to say that it can be shown that the definition that we have given allows one, inprinciple, to perform this computation.


of two probability measures. From the point of view of the algebras of random variables, this constructioncorresponds to the tensor product of algebras. Let us consider for example a probability space (Ω,P) and letus denote by (A, τ) the (commutative !) non-commutative probability space (L∞(Ω,P),E). Let us take twobounded random variables X and Y on Ω. Let us denote by (AX , τX) the subalgebra of A generated by X,that is, the set of polynomials in X, endowed with the linear form induced by the expectation. Let us define(AY , τY ) similarly from Y .

The subalgebra of A generated by X and Y , which is nothing but the space of polynomials in X and Y , isnot necessarily isomorphic to C[X,Y ] since there may be algebraic relations between X and Y . Nevertheless,this subalgebra is the image of C[X,Y ] by the natural morphism of algebras P 7→ P (X,Y ). The algebraC[X,Y ] serves as a concrete model for the tensor product AX ⊗ AY which, in a vague way, is the largestpossible algebra which is generated byAX andAY and in which each element ofAX commutes to each elementof AY . There is in particular a unique morphism of algebras m : AX ⊗AY → A such that m(X ⊗ 1) = Xand m(1⊗ Y ) = Y , and the range of this morphism is the subalgebra generated by X and Y .

One can define a state τX ⊗ τY on AX ⊗ AY by setting (τX ⊗ τY )(a ⊗ b) = τX(a)τY (b). This stepcorresponds to the construction of the tensor product of two measures. One can then, for all element c inAX ⊗ AY , compare its expectation in AX ⊗ AY and in A, or more precisely compare (τX ⊗ τY )(c) andτ(m(c)). The following proposition is the purely algebraic reformulation of independence that we were lookingfor: the variables X and Y are independent if and only if the morphism of algebras m : AX ⊗ AY → Apreserves the expectations, that is, if τX ⊗ τY = τ m. The general definition is the following.

Definition 3.3.2 Let (A, τ) be a non-commutative probability space. Let A1 and A2 be two subalgebrasof A. Let τ1 and τ2 be the restrictions of τ to A1 and A2 respectively. We say that A1 and A2 areindependent if every element of A1 commutes to every element of A2 and if the natural morphism of algebrasm : A1 ⊗A2 → A satisfies the equality τ m = τ1 ⊗ τ2.

We have gained two things with respect to the commutative case: on one hand the ambient space Aneeds not be commutative and on the other, none of the algebras A1 and A2 need to be commutative. Weonly insist that they commute to each other. This is for example the case of the two subalgebras R andP of End((CN )⊗n) defined in the context of Schur- Weyl duality (see Theorem 3.1.9). However, these twosubalgebras are not independent.

The algebraic construction which corresponds to freeness is the free product of two algebras. If A1 andA2 are two unital algebras, their free product A1 ∗A2 is the largest possible unital algebra which is generatedby A1 and A2, with the only condition that the unit of A1 coincides with the unit of A2. More precisely,a free product of A1 and A2 is an algebra A endowed with two morphisms of algebras j1 : A1 → A andj2 : A2 → A such that, an arbitrary algebra B being given, as well as two morphisms of algebras m1 : A1 → Band m2 : A2 → B, there exists a unique morphism m : A → B such that m j1 = m1 and m j2 = m2.This description characterises the triple (A, j1, j2) up to isomorphism.

A way of proving that such a free product exists consists in choosing in each of the unital algebras A1

and A2 a supplementary subspace of C: we write A1 = C⊕A01 and A2 = C⊕A0

2. Then the algebra

C⊕

A01 ⊕A0

2 ⊕(A0

1 ⊗A02

)⊕(A0

2 ⊗A01

)⊕⊕n≥3

(A0

1 ⊗A02 ⊗A0

1 ⊗ . . .)︸︷︷︸

n facteurs

⊕(A0

2 ⊗A01 ⊗A0

2 ⊗ . . .)︸︷︷︸

n facteurs

(3.6)

endowed with the most natural product (which is not absolutely straightforward to define precisely since theproduct of two elements of A0

1 for instance does not necessarily belong to A01) and the morphisms j1 and j2

which identify respectively A1 and A2 with C⊕A01 and C⊕A0

2, is a free product of A1 and A2.

Given states τ1 and τ2 on A1 and A2, one can choose A01 and A0

2 as the kernels of τ1 and τ2. A naturalstate appears then on the free product, whose kernel is the hyperplane between the brackets in (3.6). Wedenote this state by τ1 ∗ τ2 and it can be checked that A1 and A2 are free with respect to this state. We nowhave a characterisation of freeness analogous to the characterisation of independence given above.


Definition 3.3.3 Let (A, τ) be a non-commutative probability space. Let A1 and A2 be two subalgebras ofA. Let τ1 and τ2 be the restrictions of τ to A1 and A2 respectively. We say that A1 and A2 are free if thenatural morphism of algebras m : A1 ∗ A2 → A satisfies the equality τ m = τ1 ∗ τ2.

We are going to define a universal model for t-freeness by defining a state on the freeproduct of the algebras underlying two arbitrary non-commutative probability spaces.

Definition 3.3.4 Let (A1, τ1) and (A2, τ2) be two non-commutative probability spaces.Let t be a positive real. Let (U , υ) be a non-commutative probability space generated bya unitary element ut whose distribution is the measure νt on U. Let m be the uniquemorphism of algebras

m : A1 ∗ A2 → (A1 ⊗A2) ∗ Usuch that m(a1) = a1 ⊗ 1 for all a1 ∈ A1 and m(a2) = u(1⊗ a2)u−1 for all a2 ∈ A2. Wecall t-free product of τ1 and τ2 the state τ1 ∗t τ2 on A1 ∗ A2 defined by

τ1 ∗t τ2 = [(τ1 ⊗ τ2) ∗ υ] m.The fact that the measure νt is invariant by complex conjugation implies that the

pair (u, u−1) has the same distribution as the pair (u−1, u), so that replacing m by themorphism m′ such that m′(a1) = u(a1 ⊗ 1)u−1 and m′(a2) = 1 ⊗ a2 would lead to thesame definition of τ1 ∗t τ2.

For t = 0, we recover the definition of the tensor product of two states, transportedfrom the tensor product of the algebras to their free product by the natural morphismA1 ∗ A2 → A1 ⊗ A2. On the other hand, if t > 0, the element ut is not scalar in U andthis allows one to prove that m is injective. So, the subalgebra of (A1⊗A2)∗U generatedby A1 ⊗ 1 and u(1⊗A2)u−1 is a realisation of the free product of A1 and A2.

Once a universal model is defined, we can define t-freeness.

Definition 3.3.5 ([LBG8, Definition 2.5]) Let (A, τ) be a non- commutative proba-bility space. Let A1 and A2 be two subalgebras of A. Let τ1 and τ2 denote the restrictionsof τ to A1 and A2 respectively. We say that A1 and A2 are t-free if the natural morphismof algebras m : A1 ∗A2 → A satisfies the equality τ m = τ1 ∗t τ2. We say that two subsetsof A are t-free if the involutive subalgebras which they generate are.

The observation made after Definition 3.3.4 ensures that this definition is symmetricin A1 and A2.

We now check that t-convolution corresponds to t-freeness.

Proposition 3.3.6 Let (A, τ) be a non-commutative probability space. Let t ≥ 0 be a realand let a, b ∈ A be two self-adjoint elements which are t-free, with respective distributionsµ and ν. Then the distribution of a+ b is µ ∗t ν.

We can define other t-free convolutions. For example, we can multiply t-free elements:if, with the notation of the proposition above, A is a C∗-algebra and a, b are non-negative,we can define µt ν as the distribution of

√ba√b. Also, if a and b are unitary, in which

case µ and ν are measures on U, we can define µt ν as the distribution of ab.


3.3.3 Differential systems

When one tries to perform computations with pairs of t-free variables, one is led tocompute expressions of the form

τ(a1utb1u∗t . . . anutbnu

∗t ), (3.7)

where the family a1, . . . , an is independent of b1, . . . , bn and ut is free with the unionof these two families.

The best grip that one has on such expressions is to consider them as functions of tand to establish as small a differential system as possible which they satisfy. In order todifferentiate with respect to t an expression like (3.7), one uses a free stochastic differentialequation satisfied by the free unitary Brownian motion, which is the analogue (and in avague sense the limit as N tends to infinity) of the stochastic differential equation (3.1)which defines the unitary Brownian motion [Bia97].

This derivative involves products of expressions of the form (3.7). By consideringall possible products of expressions of this form where a1, . . . , an, utb1u

∗t , . . . , utbnu

∗t ap-

pear each exactly once, one obtains a finite set of functions of t which satisfies a closeddifferential system. By solving such systems, one obtains the following result.

Proposition 3.3.7 ([LBG8, Proposition 3.5]) Let us define a function G(t, z) in aneighbourhood of (0, 0) in R+ × C.

Let (A, τ) be a non-commutative probability space. Let a and b be two normal elementswhose distributions have compact support and are symmetric (by this we mean that a and−a on one hand, b and −b on the other, have the same distribution). Choose t ≥ 0.Assume that a and b are t-free. We set

G(t, z) =∑n≥1

τ((ab)2n)e2ntzn.

Then G is the unique solution, in a neighbourhood of (0, 0) in R+×C, of the non-linearequation

∂tG+ 2z∂z(G2) = 0,

G(0, z) =∑n≥1

τ(a2n)τ(b2n)zn.

This proposition allows one for example to prove (3.5). We can also use it to computethe distribution of the product of two t-free Bernoulli variables.

Proposition 3.3.8 For all t > 0, one has the following equality of measures on U:

δ−1 + δ1

2t

δ−1 + δ1

2= ρ4t(ξ

2)dξ,

where ρ4t is the density of the measure ν4t with respect to the uniform measure.


Figure 3.5: Density of δ−1+δ12 t δ−1+δ1

2 at eiθ in function of θ and t. The support of the measurefills the entire circle for the first time at t = 1. The last points to enter the support are i and−i.

3.3.4 Non-existence of t-free cumulants

By analogy with the case of freeness, we have wondered if it was possible to find t-freecumulants, that is, multilinear forms defined on every non-commutative probability spaceand which would vanish as soon as they are evaluated on a set of arguments which canbe split into two non-empty families which are t-free.

INSERT 14 – Free cumulants [NS06]

Let µ be a compactly supported probability measure on R. The classical cumulants of µ are the complexnumbers (cn(µ))n≥1 defined by the equality

log

∫Rezt µ(dt) =

∑n≥1

cn(µ)zn

n!.

Cumulants linearise the convolution: if µ and ν are compactly supported, then cn(µ ∗ ν) = cn(µ) + cn(ν) forall n ≥ 1.

Cumulants are related in a combinatorial way to moments. For all n ≥ 1, let us denote by mn(µ) =∫R t

n µ(dt) the n-th moment of µ. Let us consider the set Pn of all partitions of 1, . . . , n. For all π ∈ Pn,let us define mπ(µ) as the product of the moments of µ whose orders are the sizes of the blocks of π. Forexample, m1,3,2,5,4(µ) = m1(µ)m2(µ)2. Let us define in a similar way that cumulants cπ(µ). Thenthe relation which links moments and cumulants is the following: for all n ≥ 1,

mn(µ) =∑π∈Pn

cπ(µ).

This relation can be inverted in order to express each cumulant cn in terms of the moments mπ, by using theMobius function of the lattice Pn endowed with the partial order for which π1 ≤ π2 if each block of π1 iscontained in a block of π2. We denote by 1n = 1, . . . , n the largest element of Pn. Then

cn(µ) =∑π∈Pn

Moeb(1n, π)mπ(µ).


Let (Aτ ) be a non-commutative probability space. By analogy with what precedes, for all partition π ∈ Pnand all a ∈ A, we can define mπ(a) as the product of the moments m(ak) of a corresponding to the sizes ofthe blocks of π. In fact, if we choose a cyclic order on 1, . . . , n, for example the most natural one, we canextend this definition by setting

∀a1, . . . , an ∈ A, mπ(a1, . . . , an) =∏

B block of πB=i1<...<ir

τ(ai1 . . . air ).

The assumption of traciality that we have made on τ implies that this definition does indeed only depend onthe cyclic order of 1, . . . , n.

In order to define free cumulants, we need to introduce the lattice of non-crossing partitions.

Definition 3.3.9 Let π be a partition of the set 1, . . . , n endowed with the natural cyclic order. Toeach block B = k1, . . . , kr of π one associates the polygon PB which is the convex hull of the points

eik12πn , . . . , eik1

2πn . We say that π is non-crossing if for all B and B′ distinct blocks of π, one has PB∩PB′ = ∅.

1

2

34

5

6

7

8 9

10

1

2

34

5

6

7

8 9

10

Figure 3.6: The partition 1, 2, 3, 7, 4, 6, 5, 8, 9, 10 is non-crossing but1, 3, 7, 2, 8, 9, 10, 4, 6, 5 is crossing.

The set of non-crossing partitions of 1, . . . , n is denoted by NCn. The partial order on Pn induces apartial order on NCn which makes it a lattice. Let us denote by moeb the Mobius function of this lattice.Then the free cumulants are defined in the following way.

Definition 3.3.10 Let (A, τ) be a non-commutative probability space. Let a1, . . . , an be elements of A. Onedefines the free cumulant of a1, . . . , an by

kn(a1, . . . , an) =∑

π∈NCnmoeb(1n, π)mπ(a1, . . . , an).

Finally, for all a ∈ A and all n ≥ 1, one sets kn(a) = kn(a, . . . , a).

One has the following properties.

Proposition 3.3.11 Let (A, τ) be a probability space. Let A1 and A2 be two subalgebras of A. Then thefollowing two properties are equivalent.

1. For all n ≥ 1 and all a1, . . . , an elements of A which each belong either to A1 or to A2, but neitherall to A1 nor all to A2, that is,

a1, . . . , an ⊂ A1 ∪ A2, a1, . . . , an 6⊂ A1, a1, . . . , an 6⊂ A2,


one has kn(a1, . . . , an) = 0.2. A1 and A2 are free with respect to τ .

In particular, if a and b are two free elements of A, then one has for all n ≥ 1 the equality kn(a + b) =kn(a) + kn(b).

Thus, free cumulants linearise the free convolution and characterise freeness.

Unfortunately, in the t-free case for t > 0, we have shown that there exists nothing assimple and powerful as the free cumulants. We have looked for candidates to the role oft-free cumulants in the following set of multilinear forms.

Definition 3.3.12 Let (A, τ) be a probability space. Let n ≥ 1 be an integer. Let σ ∈ Sn

be a permutation. One defines an n-linear form on A by setting, for all a1, . . . , an ∈ A,

τσ(a1, . . . , an) =∏

(i1...ir) cycle of σ

τ(ai1 . . . air).

This definition makes sense thanks to the traciality of τ . We now look for a linearcombination of the τσ, σ ∈ Sn with the property that it vanishes as soon as it is evaluatedon arguments which can be split in two non-empty subsets which form two t-free families.

Definition 3.3.13 Let n ≥ 2 be an integer. Let t ≥ 0 be a real. A t-free cumulant of

order n is a collection (c(σ))σ∈Sn of complex numbers such that∑

σ n−cycle

c(σ) 6= 0 and such

that the following property is satisfied in every non-commutative probability space (A, τ):for all pair (A1,A2) of subalgebras of A which are t-free with respect to τ , and for alla1, . . . , an elements of A which belong each either to A1 or to A2, but neither all to A1

nor all to A2, one has ∑σ∈Sn

c(σ)τσ(m1, . . . ,mn) = 0. (3.8)

Our result is the following. By t-free for t = +∞, we mean free.

Theorem 3.3.14 ([LBG8, Theorems 4.3 and 4.4]) 1. For all t ∈ [0,+∞] and alln ∈ 2, 3, 4, 5, 6, there exists a t-free cumulant of order n. If moreover one insists thatthis cumulant is invariant by conjugation, that is such that c(σ1σ2σ

−11 ) = c(σ2) for all

σ1, σ2 ∈ Sn, then it is unique up to multiplication by a constant.2. There exists a t-free cumulant of order 7 if and only if t = 0 or t = +∞.

Perspectives

As a conclusion to these notes, I will indicate a few developments which I think arisenaturally in the continuation of the work which I have presented here, and I will do so bytaking approximately the themes in the order in which they have been treated.

After the construction of Markovian holonomy fields such as it was exposed here,it seems natural to try to complete the partial result of classification that we obtained(Theorem 1.3.6). We showed that to each regular Markovian holonomy field is associatedan admissible Levy process. Is it true that this Levy process characterises completely theMarkovian holonomy field ? In other words, is a Markovian holonomy field determined byits partition functions ? I believe that the answer is affirmative if the structure group isAbelian, but I don’t know the answer in the general case. If it turned out to be negative,which is not absolutely impossible, the best thing to do would be, in my view, to try toadd an assumption to the definition of Markovian holonomy fields which would discardthis ambiguity.

Still at the general level of the construction, I think that it would be interestingto pursue the point of view according to which a random holonomy field is a randomhomomorphism from the group of rectifiable loops on a surface and a Lie group. BenHambly and Terry Lyons have given in [HL06] a precise definition of the group of reducedrectifiable loops in a vector space or a manifold, and provided tools to handle them. Itwould probably be very difficult to understand the random variable associated to a loopby a random holonomy field as an explicit function of its signature (see [LCL] for moredetails on the signature of a path), but achieving this even in simple cases would probablyilluminate the exact regularity of Markovian holonomy fields. A perhaps less fundamentalquestion would be to study the kernel of the random homomorphism constituted by arandom holonomy field. Here and in many other questions, the case of finite structuregroups may be much simpler than the general case and yet inspiring.

A possible and natural extension of Markovian holonomy fields could be the definitionof a random holonomy along random paths, maybe Brownian paths. I think that thetechniques used in [L7] could hardly suffice to define a random holonomy along paths asirregular as typical Brownian paths, but it could be conceivable that subtle cancellationsarise due to the particular structure of the Brownian motion, allowing one to define anon-trivial object. To the best of my knowledge, only one attempt has been made in thisdirection, by Sergio Albeverio and Shigeo Kusuoka [AK95].

However, a direction of research which to my eyes is much more important is thatwhich would lead one to couple Markovian holonomy fields with other random objects.

95

96 PERSPECTIVES

These random objects could be paths, in the spirit of the last paragraph, but the physicalorigin of the objects which we consider indicates that the most interesting couplingswould probably be those with objects which could be interpreted as matter fields. Thisquestion of coupling has a sense as much in a discrete setting, on graphs, where one wouldtry to couple gauge fields with various models of statistical physics, as in a continuoussetting, where one would perhaps consider random sections of a bundle associated withthe underlying principal bundle. It is also possible that the meaningful objects are ofnon-commutative nature, thus incorporating at a more fundamental level the fermionicnature of matter.

The principle of large deviations which we have proved with James Norris concerns theYang-Mills field, which is associated with the Brownian motion on the structure group.One may ask if there are similar results for other Markovian holonomy fields, associatedwith other Levy processes. In order to establish such results, one would probably needto understand these Markovian fields as reflections of random connections, or mixtures ofrandom connections and ramified coverings. This idea is close to that exposed by Grossand Taylor in [GT93a, GT93b] where, by mixing the Yang-Mills theory and more or lessmysterious models of random coverings, they obtain remarkable formulas on the Brownianmotion in the unitary group.

One aspect of the two-dimensional Yang-Mills theory which has raised the interestof physicists is the existence, when the structure group is the unitary group U(N), ofa phase transition in the limit where N tends to infinity. This phase transition takesseveral forms depending on the point of view from which it is looked at, but I believethat it must be related to this fundamental phase transition: the limiting distribution ofthe eigenvalues of the Brownian motion on the unitary group, which for each t ≥ 0 is aprobability measure on the unit circle, is at first a Dirac mass at 1 for t = 0, then hasits support growing progressively up to time t = 4, which is the first time at which thissupport covers the whole circle, and then becomes analytic for t > 4. For t ∈ (0, 4), themoments of this limiting distribution decay like a power of their order, whereas for t > 4they decay exponentially (see Section 3.2.1). The existence of a phase transition for theYang-Mills measure on a sphere in function of the total area of the sphere [DK94] canbe made plausible in the light of these facts, but there is still much to do in order toproduce a rigorous statement. The computations on the Brownian bridge on the unitarygroup, which seem to be the natural way to attack such questions, become very quicklyintractable, even more quickly than those related to the Brownian motion. A characteristicfeature of these computations is the fact that one is constantly expressing small quantities(typically numbers of modulus less than 1) as alternated sums of huge numbers, which ofcourse is not very convenient for estimating those quantities.

The graphs presented in Figure 3.3 also illustrate the difference of behaviour of theBrownian motion on U(N) in small time and large time as N tends to infinity. In thestudy of the fluctuations of random variables of the form trf(UN), there is still much todo. We have stated a central limit theorem (Theorem 3.2.13) for functions with Lipschitzcontinuous derivative, but, with Mylene Maıda, we suspect that this theorem is true formuch less regular functions, at least when T is large enough. A more precise conjecturewould be that the class of functions for which the theorem is true is the space H

12 for

97

T > 4, and a class which depends on T for T ≤ 4. It is for instance possible thatspecial conditions should be imposed on the boundary of the spectrum of νT when T ≤ 4.However, we have no precise guess to offer about exactly what this class of functions couldbe.

With Florent Benaych-Georges, we are following our investigation of the notion of t-freeness, or of closely related notions, in particular in trying to prove theorems whichmimic the classical theorems about sums of classical independent random variables. Wehope in particular to obtain an interpolation between the Bercovici-Pata bijection andthe identical transformation of the set of probability measures on R.

A problem which mixes many of the questions which I have just mentioned, and whichI find most intriguing, it this master field which I. Singer describes informally in his paper[Sin95] and to which many physicists have devoted some effort. In the case where thespace-time is a disk, or the whole plane, all the tools required to describe this limitingfield are available, in the framework of free probability, and to prove convergence results ofthe fields associated to the groups U(N) towards this limiting field as N tends to infinity.I believe that a consistent formulation of this master field could involve the group ofreduced rectifiable loops which is a byproduct of the theory of rough paths developedby Terry Lyons [LCL], and which I have mentioned above. This group, which T. Lyonshas studied with Ben Hambly, is a kind of free group with an uncountable number ofgenerators (although it is not a free group in the algebraic sense) with which it is possibleto do analysis. In the spirit of the description of I. Singer, I believe that the master fieldcould be rigorously defined as a trace on this group and I am currently working in thisdirection.

Among the many aspects of the large N limit, I find the Makeenko-Migdal equations[MM79] particularly appealing and they appear to me as a means to understand betterthe nature of the master field. I am trying to establish them rigorously and to understandthem as a graphical translation, in the free group of the reduced loops in a graph, ofthe computation rules derived from the Ito formula applied to the Brownian motion inthe unitary group. The proof of these equations which physicists give, based on theSchwinger-Dyson equations, is both beautiful and mysterious, and does not seem to letitself be easily translated into rigorous mathematical language.

98 PERSPECTIVES

Appendix

From the Maxwell equations to theYang-Mills measure

In this appendix, I will try to sketch a path which relates the classical description of elec-tromagnetism, the one that the reader is likely to have already studied and manipulated,to a small piece of quantum field theory which gives some insight into the physical interestof the Yang-Mills measure.

A.1 Fields an potentials

The classical theory of electromagnetism, as it was formulated at the end of the nine-teenth century, consists of the four Maxwell equations and the Lorentz law. The Maxwellequations determine two time-dependent vector fields in R3, the electric field ~E and themagnetic field ~B, in terms of the density of charge ρ and the density of current ~. Theyread as follows:

div ~E = ρ, (Gauss)−→rot ~E = −∂t ~B, (Faraday)

div ~B = 0, (Thomson)−→rot ~B = ~+ ∂t ~E. (Ampere)

The Gauss equation is the fundamental law of electrostatics, the Faraday equationallows one for example to compute the current induced in a conducting spiral by a varyingmagnetic field, the Thomson law is equivalent to the non-existence of magnetic charges(or magnetic monopoles) and the Ampere law allows one for example to compute themagnetic field produced by a conducting wire traversed by an electric current.

The six components of ~E and ~B are constrained, independently of the distributionand motion of charges in space, by two homogeneous equations, namely the Faradayequation and the Thomson law. This suggests that it should be possible to expresstheir six components in terms of four scalar quantities. The Thomson law implies the

existence, at least locally, of a vector field ~A such that ~B =−→rot ~A. Such a field is called a

vector potential. Once a vector potential has been chosen, the Faraday equation impliesthe existence, at least locally, of a scalar field V , called the electric potential, such that

99

100 APPENDIX . FROM MAXWELL TO YANG-MILLS

~E = −~∇V − ∂t ~A. When one is working in R3, the potentials can be defined globally andit is possible to define ~E and ~B in terms of the four components of ~A and V .

However, the potentials are never unique. One checks easily that if ~A and V aresuitable potentials in a given situation, then, for all time-dependent function ϕ : R3 → R,the potentials ~A+ ~∇ϕ and V − ∂tϕ are just as suitable. By varying ϕ one obtains all thepairs ( ~A, V ) which describe a given situation.

A.2 The Schrodinger equation for a charged particle

The Lorentz law determines the force ~F which is exerted on a particle in function of itscharge q, its speed ~v and the electric and magnetic fields ~E and ~B:

~F = q ~E + q~v ∧ ~B. (A.1)

The first step of our progression consists in writing the Schrodinger equation for a chargedparticle exposed to a given electric and magnetic field.

In order to pass from a classical to a quantum theory, the first thing to do is toformulate the classical theory in a Lagrangian form, then in a Hamiltonian form. In ourcase, we need to find a function L (~r,~v) of two vectors of R3 such that the Euler-Lagrangeequations

d

dt

(∂L (~r,~v)

∂vi

)=∂L (~r,~v)

∂ri, i ∈ 1, 2, 3,

are equivalent to the Lorentz law (A.1). One checks directly that if a pair of potentials

( ~A, V ) has been chosen, then the function

L (~r,~v) =1

2m‖~v‖2 − qV (~r) + q~v · ~A(~r)

does this for us. In order to go from a Lagrangian formulation to a Hamiltonian one,the rule consists in defining an impulsion conjugated to ~r, denoted by ~p, defined bypi = ∂L

∂vi, i ∈ 1, 2, 3, and then in expressing the function H (~r, ~p) = ~p · ~v −L (~r,~v) in

function of ~r and ~p.In our case, we find ~p = m~v + q ~A (in contrast with the most common case where

~p = m~v), then H (~r, ~p) = 12m‖~v‖2 + qV (~r). In function of ~r and ~p, we find

H (~r, ~p) =1

2m

(~p− q ~A

)2

+ qV (~r).

With this definition of H , the Lorentz law is thus a particular case of the Hamiltonequations:

dridt

=∂H

∂pi,dpidt

= −∂H∂ri

, i ∈ 1, 2, 3.

For us, the main interest of this formulation is that it allows us to write a Schrodingerequation in an automatic way. Indeed, we are going to define a linear operator on theHilbert space L2(R3) by replacing, in the expression of H , each scalar function of ~r by

A.3. THE EFFECT OF A CHANGE OF GAUGE ON THE WAVE FUNCTION 101

the operator of multiplication by this function, and each component of ~p by −i times thepartial derivation in the corresponding direction. In this way, we find the Hamiltonianoperator

H = − 1

2m

(~∇− iq ~A

)2

+ qV.

The Schrodinger equation writes then simply

∂tΨ = −iHΨ,

where Ψ is the wave function of a particle with charge q.

A.3 The effect of a change of gauge on the wave function

It is a pleasant fact that the Hamiltonian which we have just written depends on thephysical data ~E and ~B through the potentials V and ~A. Indeed, these potential encodethe same information contained in ~E and ~B in a less redundant way. However, we havealready mentioned that these potentials are not unique. Choosing a particular pair ( ~A, V )is called choosing a gauge and we are going to see that this is indeed the same gauge asthe one mentioned in gauge theories, of which the Yang-Mills theory is an example.

Let us thus pick a gauge, as we did earlier, by choosing V and ~A. For all time-dependent function ϕ : R3 → R, let Hϕ denote the Hamiltonian obtained by replacing ~A

by ~A+ ~∇ϕ and V by V − ∂tϕ. For all ϕ, the Hamiltonians H and Hϕ describe the samephysical situation, we could have obtained one as well as the other by the procedure whichwe have followed. As one could expect, there is a simple transformation which from awave function satisfying the Schrodinger equation for H makes a wave function satisfyingthe Schrodinger equation for Hϕ. In order to determine this transformation, we observethat

eiqϕ(~∇− iq ~A)2e−iqϕ =(~∇− iq( ~A+ ~∇ϕ)

)2

,

from which we extract the relation

Hϕ = eiqϕHe−iqϕ − q∂tV,

and the equivalence

∂tΨ = −iHΨ⇐⇒ ∂t(eiqϕΨ

)= −iHϕ

(eiqϕΨ

).

We can thus describe the same physical situation by two wave functions which differ bya phase which depends on the time and the point where one stands. Of course, the physicsis the same, since these two descriptions are associated with two distinct Hamiltonians.Moreover, if we add a second particle to our system, its wave function will be affected bythe same transformation as that of the first particle when passing from one Hamiltonianto the other. So, possible interferences between the wave functions which would haveobservable consequences will be identical in the two descriptions.


A.4 The geometrical nature of the vector potential

For the sake of simplicity, let us work in a situation where the electric field is zero andthe magnetic field is constant. We choose V = 0. Even under these restrictive hypothese,we have just seen that a huge group acts on the set of possible ways of describing thephysical situation which we consider: the group of all functions on R3 with values in thegroup U(1) of complex numbers of modulus 1. This group is called the gauge group – wenow know why – and we denote it by J . The product in J of two elements eiϕ and eiϕ

′is

simply ei(ϕ+ϕ′). An element eiϕ of J acts on the wave function Ψ of a particle of charge qaccording to the formula (eiϕ ·Ψ)(~r) = eiqϕ(~r)Ψ(~r). It also transforms the vector potential~A into ~A+ ~∇ϕ, and the differential operator ~∇− iq ~A into ~∇− iq( ~A+ ~∇ϕ).

In order to go from the multitude of descriptions to which we are confronted to anintrinsic description, which requires no particular choice of a gauge, we need to ground ourdescription on an object whose symmetry group is the gauge group, that is, a principalbundle. In the present situation, we need to consider the space R3 × U(1) endowed withthe action, denoted on the right, of U(1) given by (~r, eiθ) · eiα = (~r, ei(θ+α)). This spaceis fibred by the projection on the first component R3 × U(1) → R3. An automorphismof this principal bundle is by definition a diffeomorphism of R3 × U(1) which commutesto the action of U(1) and preserves the fibres, that is, which sends all pair (~r, eiθ) on apair of the form (~r, eiθ

′). It is not difficult to see that the group of automorphisms of the

principal bundle R3 × U(1) identifies with J : an element eiϕ of J acts on a pair (~r, eiθ)according to eiϕ · (~r, eiθ) = (~r, ei(ϕ(~r)+θ)).

We claim now that the wave function of a particle of charge q is in fact a functionψ : R3×U(1)→ C which is q-equivariant in the following sense: for all (~r, eiθ) ∈ R3×U(1)and all eiα ∈ U(1), one has

ψ((~r, eiθ) · eiα) = ψ(~r, ei(θ+α)) = eiqαψ(~r, eiθ).

Each usual wave function, one of those which we have denote by Ψ, is just the function ψread through a particular section of the bundle R3×U(1) : for each function σ : R3 → U(1),called section of the bundle, we can define

Ψσ(~r) = ψ(~r, σ(~r)).

When the section σ is replaced by a new section σ′, the latter can be written under theform eiϕσ for a certain eiϕ ∈ J and one finds the following relation between Ψσ and Ψσ′ :

Ψσ′(~r) = ψ(~r, eiϕ(~r)σ(~r)) = eiqϕ(~r)Ψσ(~r).

Changing the section through which one reads the wave function produces thus the sameeffect on the wave function as changing the vector potential.

We now need to discuss the nature of ~A in this new geometrical context. In theSchrodinger equation, the field ~A appears as the 0 order term in the differential operator~∇− iq ~A. In order to understand this term, we are going to see how the derivatives of ψcan be expressed in terms of those of Ψσ.

Let thus ψ be our wave function and let ~r be a point of R3. Let us choose a sectionσ : R3 → U(1) and set σ(~r) = eiθ. A tangent vector to R3×U(1) at (~r, eiθ) is a pair (~x, iξ)

A.5. A LAGRANGIAN FOR THE YANG-MILLS EQUATIONS 103

belonging to R3⊕ iR. Let us choose such a vector (~x, iξ) and compute the differential of ψat (~r, eiθ) in the direction (~x, iξ) in terms of Ψσ. Writing, for all ~s ∈ R3, σ(~s) = ei(θ+ϕ(~s)),with ϕ(~r) = 0, we find

d(~r,eiθ)ψ(~x, iξ) =d

dt |t=0ψ(~r + t~x, ei(θ+tξ))

= ~∇Ψσ(~r) · ~x− iq(−ξ + ~∇ϕ(~r) · ~x)Ψσ(~r).

We recognise a familiar expression. Indeed, if we constrain ~x and ξ by the relationξ = − ~A · ~x in the last expression, we find

(~∇− iq( ~A+ ~∇ϕ))Ψσ(~r) · ~x = d(~r,σ(~r))ψ(~x, i ~A · ~x).

In what precedes, we have implicitly used the constant section equal to eiθ as reference.Any other section writes eiϕeiθ for some eiϕ ∈ J . In fact, the choice of θ is harmless, wecould have taken θ = 0, and we have thus obtained the relation

(~∇− iq( ~A+ ~∇ϕ))Ψeiϕ(~r) · ~x = d(~r,eiϕ(~r))ψ(~x, i ~A · ~x).

Thus, applying the differential operator ~∇− iq( ~A+ ~∇ϕ) to Ψeiϕ – both depending onthe gauge – amounts to differentiating ψ in a certain direction which does not depend onthe gauge, but only on ~A. In fact, ~A determines, at each point ~r of R3 and for all eiθ

of U(1), a linear mapping from the tangent space to R3 at ~r into the tangent space toR3 × U(1) at (~r, eiθ) :

~x 7→ (~x, i ~A · ~x) ∈ R3 ⊕ iR.The range of this linear mapping is called the horizontal subspace associated with ~A. Thecollection of all horizontal subspaces determined by ~A is a connection. At each point ofR3×U(1), they indicate a way of identifying – of connecting – infinitesimally close fibres.

The differential operator which in the gauge eiϕ reads ~∇ − iq( ~A + ~∇ϕ) is the covariantgradient associated with the connection. A wave function ψ has a vanishing covariantgradient at a point if at this point, its differential is zero in all horizontal directions.

Finally, our mathematical description of the situation is the following: the wave func-tion of a particle of charge q is a q-equivariant function on R3 × U(1) and the magneticfield determines a connection on R3 ×U(1), that is, the data at each point of a subspaceof the tangent space which one calls the horizontal subspace. If a section σ : R3 → U(1)is given, then one can characterise the horizontal distribution at each point by comparingit with the differential of the mapping (id, σ) : R3 → R3 × U(1). The difference is apurely imaginary number, which at a given point of R3 depends linearly on the directionin which it is computed. It is thus a differential 1-form, that which in the reference gaugewas written above as ~x 7→ ~A · ~x.

A.5 A Lagrangian for the Yang-Mills equations

We have now elucidated the geometrical nature of the vector potential ~A. In a completelyquantum theory of electromagnetism, not only the particles are being treated in a quantum


way, but also the electromagnetic field itself. In a caricatural way, one can say that, justas a particle, classically represented at each time by a point ~r in R3, is represented ina quantum theory by a wave function depending on time on R3, the electromagneticfield, classically represented by a function ( ~E, ~B) from R3 to R6, is replaced by a “wavefunctional” on the space of these functions. And just like for particles, this quantificationrequires that one writes the Maxwell equations in a Lagrangian form.

In order to do so, we need first to rewrite the Maxwell equations under a form whichemphasises better their geometric structure. Without even referring to the discussionabove where we have analysed the structure of ~A, simple experiments where a physicalinstallation is compared with its image in a mirror or by a homothety indicate thatneither ~B nor ~E transform like vectors. They transform in fact like differential forms.More precisely, it will be soon apparent that it is very convenient to define a 1-form Eand a 2-form B by setting

E = Exdx+ Eydy + Ezdz and B = Bxdy ∧ dz +Bydz ∧ dx+Bzdx ∧ dy.

One then combines these two forms into one single 2-form F called electromagnetic fieldby setting

F = B − dt ∧ E.Finally, one encodes the density of charge ρ and the density of current ~ by a differentialform, setting

J = −ρdt+ jxdx+ jydy + jzdz.

If we endow R4 with the metric −dt2 + dx2 + dy2 + dz2, we define a Hodge operator ondifferential forms, denoted by ∗. It differs from the operator associated with the Euclideanmetric only with an extra minus sign when it is applied to a form which involves dt.

The Maxwell equations can then be rewritten in very few symbols:dF = 0, (Gauss, Thomson)d(∗F ) = ∗J . (Faraday, Ampere)

The homogeneous equation implies the existence of a 1-form A such that F = dA.With our previous notation, one can take A = −V dt+ Axdx+ Aydy + Azdz.

This formulation of the Maxwell equations has the huge advantage of making ittransparent that they are invariant under the transformations of R4 which preserve theMinkowski metric. Thus, the theory of electromagnetism as Maxwell had formulated itwas already a relativistic theory and it is indeed some of its consequences which have ledEinstein to formulate the theory of special relativity.

For us on the other hand who would like to quantify the electromagnetic field, aformulation such as this one which does not anymore distinguish clearly between spaceand time is slightly uncomfortable. We are simply going to check that the Lagrangian

L (A) =1

2F ∧ ∗F + A ∧ ∗J

is a Lagrangian for the electromagnetic field in presence of the charges and currentsdescribed by J .

A.6. PATH INTEGRALS 105

First of all, this Lagrangian comes out as a function of the 1-form A rather than ofF , and the relation F = dA is implicit, which immediately implies dF = 0. Disregardingthe issue of boundary conditions, we need now to determine the 1-forms A for which theintegral over R4 of the Lagrangian is stationary. If we vary A by δA, then F varies bydδA. Using the fact that, for arbitrary 2-forms α and β, one has α ∧ (∗β) = (∗α)∧ β, wefind, at the first order,

L (A+ δA) = L (A) +

∫R4

(d ∗ F − ∗J) ∧ δA.

Since the pairing (α, β) 7→∫R4 α ∧ β is non- degenerate, it follows that the integral of the

Lagrangian is stationary if and only if d ∗ F = ∗J .

A.6 Path integrals

With a Lagrangian formulation of the theory of electromagnetism in empty space, wecan apply Feynman’s method of quantification by path integrals. In this method, oneassigns to each configuration A the integral of its Lagrangian S(A) =

∫R4 L (A) and

then a probability amplitude ei~S(A). In order to determine the probability that a certain

phenomenon occurs, one adds up all the amplitudes of the configurations of A which couldproduce it and one computes the square of the modulus of the result. One thus arrivesat integrals of the form ∫

BeiS(A) dA,

where B is the space of configurations corresponding to the phenomenon of which we wantto compute the probability.

We have arrived at a point where integrals have appeared on sets of connections, withan exponential weight involving the Lagrangian of electromagnetism, that is, almost thepoint at which the introduction of these notes started. Replacing the Minkowski metricon R4 by the Euclidean metric, the integral of the Lagrangian becomes a positive functionon the space of connections. Replacing e

i~S(A) by e−

12S(A) and working in empty space,

where J = 0, one finds the integral ∫Be−

12S(A) dA,

which is a variant of the heuristic expressions which we have given for the Yang-Millsmeasure (see (2) and (2.1)).

List of works

[L1] Thierry Levy. Construction et etude a l’echelle microscopique de la mesure deYang-Mills sur les surfaces compactes. C. R. Acad. Sci. Paris Ser. I Math.,330(11):1019–1024, 2000.

[L2] Thierry Levy. Yang-Mills measure on compact surfaces. Mem. Amer. Math. Soc.,166(790):xiv+122, 2003.

[L3] Thierry Levy. Wilson loops in the light of spin networks. J. Geom. Phys.,52(4):382–397, 2004.

[L4] Thierry Levy. Discrete and continuous Yang-Mills measure for non-trivial bundlesover compact surfaces. Probab. Theory Related Fields, 136(2):171–202, 2006.

[LN5] Thierry Levy and James R. Norris. Large deviations for the Yang-Mills measureon a compact surface. Comm. Math. Phys., 261(2):405–450, 2006.

[L6] Thierry Levy. Schur-Weyl duality and the heat kernel measure on the unitarygroup. Adv. Math., 218(2):537–575, 2008.

[L7] Thierry Levy. Two-dimensional Markovian holonomy fields. vi+166, paratredans Asterisque. [arXiv 0804.2230]

[LBG8] Florent Benaych-Georges and Thierry Levy. A continuous semigroup of notions ofindependence between the classical and the free one. Prpublication, 2009. [arXiv0811.2335]

[LM9] Thierry Levy and Mylene Maıda. Central limit theorem for the heat kernelmeasure on the unitary group. Prpublication, 2009.

Seminars and proceedings

[A1] Thierry Levy. Comment choisir une connexion au hasard? In Seminaire deTheorie Spectrale et Geometrie. Vol. 21. Annee 2002–2003, volume 21 du Semin.Theor. Spectr. Geom., pages 61–73. Univ. Grenoble I, Saint, 2003.

[A2] Thierry Levy. Wilson loops and spin networks. In XIVth International Congresson Mathematical Physics, pages 498–504. World Sci. Publ., Hackensack, NJ, 2005.

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108 LIST OF WORKS

[A3] Thierry Levy. Large deviations for the two-dimensional Yang-Mills measure.In Stochastic analysis in mathematical physics, pages 54–68. World Sci. Publ.,Hackensack, NJ, 2008.

Lecture notes

[LCL] Terry J. Lyons, Michael Caruana, and Thierry Levy. Differential equations drivenby rough paths, volume 1908 of Lecture Notes in Mathematics. Springer, Berlin,2007. Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, July 6–24, 2004, With an introduction concerning the Summer School byJean Picard.

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Contents

Introduction 3Presentation of the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

The mathematical difficulties of gauge theories . . . . . . . . . . . . . . . . . . . . . . . . 4The construction of the Yang-Mills measure . . . . . . . . . . . . . . . . . . . . . . . . . . 7Large unitary matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Markovian holonomy fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Localisation of the main statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1 Markovian holonomy fields [L2, L3, L7] 131.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.1.1 The Poisson process modulo n indexed by loops . . . . . . . . . . . . . . . . . . . 131.1.2 A first definition of Markovian holonomy fields . . . . . . . . . . . . . . . . . . . . 16

E1. Topology of surfaces [Mas77, Moi77] . . . . . . . . . . . . . . . . . . . . . . . . . 161.1.3 Transition kernels and surgery of surfaces . . . . . . . . . . . . . . . . . . . . . . . 17

1.2 Multiplicative processes indexed by paths . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.2.1 The measureable space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

E2. Compact Lie groups [BtD85, DK00] . . . . . . . . . . . . . . . . . . . . . . . . . 221.2.2 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.2.3 Levy processes and marginals associated with graphs . . . . . . . . . . . . . . . . . 24

E3. Levy processes on a compact Lie group[Lia04] . . . . . . . . . . . . . . . . . . . . 241.2.4 Gauge symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.2.5 A random representation of the group of loops . . . . . . . . . . . . . . . . . . . . 28

E4. Group of loops in a graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.2.6 Wilson loops [L3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.2.7 Invariance by subdivision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.2.8 Extension to rectifiable loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

E5. Topology on the space of paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34E6. The Banchoff-Pohl inequality [BP72, Vog81] . . . . . . . . . . . . . . . . . . . . . 35

1.3 Markovian holonomy fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361.3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361.3.2 Existence and partial classification . . . . . . . . . . . . . . . . . . . . . . . . . . . 381.3.3 The Yang-Mills field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401.3.4 Link with topological quantum field theories . . . . . . . . . . . . . . . . . . . . . 42

E7. Topological quantum field theories [Ati88] . . . . . . . . . . . . . . . . . . . . . . 42

2 Holonomy fields, coverings and connections 472.1 Finite groups and ramified coverings [L7, Chapter 5] . . . . . . . . . . . . . . . . . . . . . 47

2.1.1 Fundamental group and coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.1.2 Random ramified coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.2 Brownian holonomy field and random connections [L4, LN5] . . . . . . . . . . . . . . . . . 51

115

116 CONTENTS

2.2.1 A random differential 1-form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.2.2 Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.3 Yang-Mills action and measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.4 A large deviations principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.4.1 Sobolev spaces of connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58E8. Schilder’s theorem [DZ93] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.4.2 The rate functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60E9. Classification of principal G-bundles . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.4.3 The Brownian holonomy field and its disintegration . . . . . . . . . . . . . . . . . 63E10. Brownian motion on a compact connected Lie group . . . . . . . . . . . . . . . . 63

2.4.4 The large deviation principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3 Unitary Brownian motion 693.1 Combinatorial aspects of the unitary Brownian motion [L6] . . . . . . . . . . . . . . . . . 69

3.1.1 Definition of the unitary Brownian motion . . . . . . . . . . . . . . . . . . . . . . . 693.1.2 Distribution of the eigenvalues and Newton sums . . . . . . . . . . . . . . . . . . . 713.1.3 Combinatorial expression of the moments of the spectral measure . . . . . . . . . . 723.1.4 Enumeration of paths in the symmetric group . . . . . . . . . . . . . . . . . . . . . 743.1.5 Unitary Brownian motion and symmetric random walk . . . . . . . . . . . . . . . 75

E11. Schur-Weyl duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.1.6 Random ramified coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.2 Asymptotic behaviour of the eigenvalues [LM9] . . . . . . . . . . . . . . . . . . . . . . . . 793.2.1 Distribution of the eigenvalues : asymptotic results . . . . . . . . . . . . . . . . . . 79

E12. Free probability and free unitary Brownian motion [VDN92, ?, Bia97] . . . . . . . 813.3 Between independence and freeness [LBG8] . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.3.1 Convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863.3.2 Dependence structures and t-freeness . . . . . . . . . . . . . . . . . . . . . . . . . . 87

E13. Dependence structures and universal models . . . . . . . . . . . . . . . . . . . . 873.3.3 Differential systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903.3.4 Non-existence of t-free cumulants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

E14. Free cumulants [NS06] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Perspectives 95

From Maxwell to Yang-Mills 99A.1 Fields an potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99A.2 The Schrodinger equation for a charged particle . . . . . . . . . . . . . . . . . . . . . . . . 100A.3 The effect of a change of gauge on the wave function . . . . . . . . . . . . . . . . . . . . . 101A.4 The geometrical nature of the vector potential . . . . . . . . . . . . . . . . . . . . . . . . 102A.5 A Lagrangian for the Yang-Mills equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 103A.6 Path integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

random holonomy, large unitary matrices · after ambar sengupta had given the rst one. since then,...

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