values of brownian intersection exponents, i: half-plane ... · exponents from the sle6-exponents,...

I. Benjamini, O. Häggström (eds.), Selected Works of Oded Schramm, Selected Works in Probabilityand Statistics, DOI 10.1007/978-1-4419-9675-6_28, C© Springer Science+Business Media, LLC 2011

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Acta Math., 187 (2001), 237-273 © 2001 by Institut Mittag-Leffler. All rights reserved

Values of Brownian intersection exponents, I: Half-plane exponents

GREGORY F. LAWLER

Duke University Durham, NC, U.S.A.

by

ODED SCHRAMM and WENDELIN WERNER

Microsoft Research Redmond, WA, U.S.A.

and

The Weizmann Institute of Science Rehovot, Israel

1. Introduction

Universite Paris-Sud Or-say, France

Theoretical physics predicts that conformal invariance plays a crucial role in the macro

scopic behavior of a wide class of two-dimensional models in statistical physics (see, e.g.,

[4], [6]). For instance, by making the assumption that critical planar percolation behaves

in a conformally invariant way in the scaling limit, and using ideas involving confor

mal field theory, Cardy [7] produced an exact formula for the limit, as N -+00, of the

probability that, in two-dimensional critical percolation, there exists a cluster crossing

the rectangle [0, aN] x [0, bN]. Also, Dllplantier and Saleur [13] predicted the "fractal

dimension" of the hull of a very large percolation cluster. These are just two examples

among many such predictions.

In 1988, Duplantier and K won [12] suggested that the ideas of conformal field theory

can also be applied to predict the intersection exponents between random walks in Z2

(and Brownian motions in R2). They predicted, for instance, that if Band B' are

independent planar Brownian motions (or simple random walks in Z2) started from

distinct points in the upper half-plane H={(x,y):y>O}={zEC:Im(z»O}, then when

n-+oo,

P[B[O, n] nB'[O, n] = 0] = n-(+o(l) (1.1)

and

P[B[O, n]nB'[O,n] = 0 and B[O, n]UB'[O,n] cH] =n-(+O(l), (1.2)

where

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238 G. F. LAWLER, O. SCHRAMM AND W. WERNER

Very recently, Duplantier [11] gave another physical derivation of these conjectures based

on "quantum gravity" .

In 1982, Mandelbrot [35] suggested that the Hausdorff dimension of the Brownian

frontier (i.e., the boundary of a connected component of the complement of the path)

is 1, based on simulations and the analogy with the conjectured value for the fractal

dimension of self-avoiding walks predicted by Nienhuis (also ~; see, e.g., [33]).

To date, none of the physicists' arguments have been made rigorous, and it seems

very difficult to use their methods to produce proofs. Very recently, Kenyon [16], [17], [18]

managed to derive the exact values of critical exponents for "loop-erased random walk"

that theoretical physicists had predicted (Majumdar [34], Duplantier [10]). Kenyon's

methods involve the relation of the loop-erased walk to the uniform spanning tree and to

domino tilings. Kenyon shows that the equations relating probabilities of some domino

tiling events are discrete analogues of the Cauchy-Riemann equations, and therefore

the probabilities can be approximated by analytic functions with prescribed boundary

behavior. These methods do not seem applicable for the goals of the present paper.

For planar Brownian motions, it is easy to show, using subadditivity arguments and

the scaling property, that there exist positive finite numbers C and ( such that (1.1) and

(1.2) are true. Up to the present paper, there was not even a mathematical heuristic

arguing that the values of ( and ( are i and ~. Burdzy-Lawler [5J (see also [9], [24])

showed that the intersection exponents were indeed the same for simple random walks as

for Brownian motions; Lawler [21] proved that the Hausdorff dimension of the set of cut

points of a Brownian path is 2 - 2(. He also showed (see [22], [23]) that the dimension

of the Brownian frontier (and more generally the whole multifractal spectrum of the

Brownian frontier) can be expressed in terms of exponents defined analogously to C. As

part of that work, he showed that the right-hand side of (1.1) can be replaced with

n-c'g(n) where 9 is bounded away from 0 and infinity; we expect that the argument can

be adapted to show that the same is true for (1.2).

Recently, Lawler and Werner [28] extended the definition of intersection exponents

in a natural way to "non-integer packets of Brownian motions", and derived certain

functional relations between these exponents. These relations indicate that Mandelbrot's

conjecture that the dimension of the Brownian frontier is ~ is indeed compatible with the

predictions of Duplantier-K won. It turned out that intersection exponents in the half

plane play an important role in understanding exponents in the whole plane. Conformal

invariance of planar Brownian motion is a crucial tool in the derivation of these relations.

In particular, there is a measure on Brownian excursions in domains that has some strong

conformal invariance properties, including a "restriction" (or "locality") property.

In another paper, Lawler and Werner [29J showed that intersection exponents as-

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VALUES OF BROWNIAN INTERSECTION EXPONENTS, I: HALF-PLANE EXPONENTS 239

sociated to any conformally invariant measure on sets with this restriction property are

very closely related to the Brownian exponents. This provides a rigorous justification to

the link between the conjectures regarding intersection exponents for planar Brownian

motions and conjectures for intersection exponents of critical percolation clusters (see

[13], [8], [3]), because percolation clusters are conjectured to be conformally invariant in

the scaling limit-see, e.g., [19], [2]-and they should also have a restriction property

(because of the independence properties of percolation). The question of how to compute

these exponents remained open.

Independently, Schramm [42] defined a new class of conformally invariant stochastic

processes indexed by a real parameter K::;::a, called SLE", (for stochastic L6wner evolu

tion process with parameter K:). The definition of these processes is based on L6wner's

ordinary differential equation that encodes in a conformally invariant way a continuous

family of shrinking domains (see, e.g., [32], [37]). More precisely, [42] defines a family of

conformal maps gt from subsets D t of H onto H by the equation -2

8t gt (z) = (3 ( )' "t-gt Z

(1.3)

where (3 is a standard Brownian motion on the real line. (Actually, in [42], instead

of (1.3), the corresponding equation for the inverse maps g;1 is considered.) The do

main D t can be defined as the set of zoEH such that a solution gs(zo) of this equation

exists for sE [a, t]. When t increases, the set Kt=H\Dt increases: Loosely speaking,

(Kt, t:;::O) can be viewed as a growing "hull" that is penetrating the half-plane. By

applying a conformal homeomorphism f: H-+D, SLE", can similarly be defined in any

simply-connected domain DC;C.

In [42], the main focus is on the case K:=2, which is conjectured there to correspond

to the scaling limit of loop-erased random walks, but the conjecture that SLE6 corre

sponds to the scaling limit of critical percolation cluster boundaries is also mentioned.

In particular, it is possible to compute explicitly the probability that an SLE6 crosses a

rectangle of size a x b. It turns out that this result is exactly Cardy's formula. This gives

a mathematical proof for Cardy's formula, assuming the still open conjecture that SLE6

is indeed the scaling limit of percolation cluster boundaries.

The main goal of the present paper is to prove some of the conjectured values of

intersection exponents of Brownian motion in a half-plane.

THEOREM 1.1. Let Bl, ... , BP denote p independent planar Brownian motions (p:;::2)

started from distinct points in the upper half-plane H. Then, when t-+oo,

P[lfi#jE{I, ... ,p}, Bi[a,t]nBi[a,t] =0 and Bi[O,t]cH]=r(p+O(l),

where

(p = ~p(2p+l).

862

240 G. F. LAWLER. O. SCHRAMM AND W. WERNER

These values have been predicted by Duplantier and Kwon [12]. In particular, C= - 5 (2="3'

We also establish the exact value (and confirm some of the conjectures stated in

[28], [11]) of more general intersection exponents between packets of Brownian motions

in the half-plane; see Theorem 4.1.

The proof of Theorem 1.1 uses a combination of ideas from the papers [28]' [29], [42]. However, to make the paper more accessible and self-contained, we attempt to review

and explain all the necessary background. The reader who wishes to see complete proofs

for all stated theorems has to be familiar with the basics of stochastic calculus and con

formal mapping theory, and read about the excursion measure and the cascade relations

from [281.

Although, at present, a proof of the conjecture that SLE6 is the scaling limit of

critical percolation cluster boundaries seems out of reach, this conjecture does lead one

to believe that SLE6 must satisfy a "locality" property; namely, it is not affected by the

boundary of a domain when it is in the interior. This locality property for SLE6 is stated

more precisely and proved in §2. It is worthwhile to note that the locality property does

not hold for the SLE",-processes when /'i, #6. In §3, we prove that SLE6 satisfies a generalization of Cardy's formula for percola

tion crossings probabilities. From this, exponents associated with the SLE6-process are

computed.

In §4, universality ideas from [29] are used to compute the half-plane Brownian

exponents from the SLE6-exponents, which completes the proof of Theorem 1.1.

In a final short §5, the conjectured relationship between SLE6 and critical percolation

is discussed. It is demonstrated that this conjecture implies a formula from the physics

literature [13], [8], [31 for the exponents corresponding to the event that there are k

disjoint percolation crossings of a long rectangle.

In the subsequent papers [25], [26], [27], we determine the exponents in the full

plane and the remaining half-plane exponents. In particular, we prove that (=~, and

also establish Mandelbrot's conjecture that the Hausdorff dimension of the frontier of

planar Brownian motion is ~.

It might be worthwhile to explain why the Brownian intersection exponents are

accessible through SLE6 , but are difficult to compute directly. In a way, the SLE6-

process is simpler, since K t continuously grows from its outer boundary. This meanS

that when studying its evolution, one can essentially forget its interior, and only keep

track of the exterior of K t . By conformal invariance, this reduces problems to finitely

many dimensions. The situation with planar Brownian motion is completely different,

since it may enter holes it has surrounded and emerge to the exterior someplace else.

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Many computations with SLE", are readily convertible to PDE problems, and in the

presence of enough symmetry, some variables can often be eliminated, converting the

PDE to an ODE.

2. SLE6 and its locality property

2.1. The definition of chordal SLEI<. and some basic properties

Let «(3t, t;::O) be a standard real-valued Brownian motion starting at ;30=0, let /'\;>0, and

let Wt=(3"'t. Consider the ordinary differential equation

(2.1)

with go(z)=z. For every zoEH and every T>O, either there is a solution of (2.1) for

tE [0, T] and for all z in a neighborhood of zo, or there is some toE(O, T] such that

the solution exists for tE [0, to) and limVto gt(z)= Wt~. Let DT be the (open) set of

zEH such that the former is true, and let KT be the set of zEH such that the latter

holds. By considering the inverse flow 8tGt(z)=2(Wr_t-Gt(z))-1, it is easy to see that

gt(Dt}=H, and that gt: Dt-+H is conformal. The process gt, t;;::O, will be called the

chordal stochastic Lovmer evolution process with parameter /'\;, or just SLE",; see [42].

In [42], a variation of this process, which we now call mdial SLE", was also studied. In

the current paper, we will not use radial SLE"" and therefore the word "chordal" will

usually be omitted. (However, radial SLE", plays a major role in a subsequent paper [25].)

The set Kt=H\Dt will be called the hull of the SLE. The process Wt will be called the

driving process of the SLE.

It is easy to verify that each of the maps gt satisfies the hydrodynamic normalization

at infinity:

lim g(z)-z=O. z ..... oo

(2.2)

Remarks. It will be shown [40] that for all /'\; #8 the hull K t of SLE", is a.s. generated

by a path. More precisely, a.s. the map 'Y( t):= gil (Wt) is a well-defined continuous path

in II, and for every t;::O the domain Dt is the unbounded connected component of

H\ "1([0, tl). There, it will also be shown that when /'\;~4 a.s. K t is a simple path for all

t>O. This is not the case when /'\;>4 [42]. However, these results will not be needed for

the current paper or for [25], [26], [27].

Lowner [32] considered the equation

864

242 C. F. LAWLER, O. SCHRAMM AND W. WERNER

with 90(Z)=Z, where z is in the unit disk, and ((t) is a parameter. He used this equation

in the study of extremal problems for classes of normalized conformal mappings. In

L6wner's differential equation, the maps 9t satisfies 9t(O)=0. The equation (2.1) is an

analogue of L6wner's equation in the half-plane, where the boundary point 00 is fixed

instead of 0, and (( t) is chosen to be scaled Brownian motion.

Marshall and Rohde [36] study conditions on ((t) which imply that K t is a simple

path.

We now note some basic properties of SLEII:'

PROPOSITION 2.1. (i) [Scaling] SLEII: is scale-invariant in the following sense. Let

K t be the hull of SLEII:' and let a>O. Then the process tf--ta- 1/ 2 Kat has the same law

as tf--tKt .

(ii) [Stationarity] Let gt be an SLEII:-process in H, driven by Wt", and let T be any

stopping time. Set.9t (z) = gT+t 09; 1 (z + W;) - W; . Then fit is an SLEII: -process in H

starting at 0, which is independent from {gt: tE [0, T]}.

Proof. (i) If K t is driven by Wt", then a- 1/ 2K,:xt is driven by a-l/2W~t, which has

the same law as Wt".

(ii) The process .9t is driven by Wt+T - W;. o

We now consider the definition of SLEII: in domains other than H. Let f: D -+ H be a conformal homeomorphism from some simply-connected do

main D. Let ft be the solution of (2.1) with fo{z)=f{z). Then (ft, t;;::O) will be called

the SLEII: in D starting at f. If gt is the solution of (2.1) with 9o{z)=Z, then we have

ft=gtOf. If K t is the hull associated to gt, then the hull associated with ft is just f-l(Kt ).

Suppose that aD is a Jordan curve in C, and let a, bEaD be distinct. Then we may

find such an f: D-+H with f(a)=O and J(b)=oo. Let K! be the SLEII:-hull associated

with the SLEII: -process starting at f. If J*: D -+ H is another such map with J* (a) = 0

and J*(b)=oo, then J*(z)=aJ(z) for some a>O. By Proposition 2.1, the corresponding

SLE",-hull K{ has the same law as a linear time change of K{ This makes it natural to

consider K! as a process from a to b in D, and to ignore the role of f. However, when

D is not a Jordan curve, some care may be needed since the conformal map f does not

necessarily extend continuously to the boundary. Partly for that reason, we have chosen

to stress the importance of the conformal parameterization f.

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2.2. The locality property

The main result of this section can be loosely described as follows: an SLE6 process

does not feel where the boundary of the domain lies as long as it does not hit it. This is

consistent with the conjecture [42] that the SLE6-process is the scaling limit of percolation

cluster boundaries, which is explained in §5. This restriction property can therefore be

viewed as additional evidence in favor of this conjecture. This feature is special to SLE6 ;

it is not shared by SLE~ when ",#6.

Such properties were studied in [29] and called "complete conformal invariance"

(when combined with a conformal invariance property). As pointed out there, all pro

cesses with complete conformal invariance have closely related intersection exponents.

Let us first state a general local version of this result. We will say that the path

"( is nice if it is a continuous simple path T [0, l]-+H such that "((0), "((l)ER\{O} and

,,((O,l)CH. We then call the connected component N =N("() of H\,,([O, 1] such that

OEaN a nice neighborhood of 0 in H. Note that N can be bounded or unbounded,

depending on the sign of "(0)"(1). When N is a nice neighborhood of 0, one can find

a conformal homeomorphism 1/J=1/JN from N onto H such that 1/J(0) =0, 1/J'(0)=1 and

1/J-l(oo) is equal to 00 if N is unbounded and to "((1) if N is bounded.

THEOREM 2.2 (locality). Let f: D-+H be a conformal homeomorphism from a

domain DcC onto H. Suppose that N is a nice neighborhood of 0 in H. Define

D*=f-l(N) and let f* be the conformal homeomorphism 1/JN o f from D* onto H. Let

KtcD be the hull of SLE6 starting at f, and let r:=sup{t:Kt naD*nD=0}. Let K;

denote SLE6 in D* started at 1*, and let r*:=sup{t:K;naD*nD=0}.

Then the law of (Kt, t<r) is that of a time change of (K;, t<r*).

Note that in this theorem, we have not made any regularity assumption on the

boundary of the domain D.

A consequence of this result is that, modulo time change, one can define the hull of

SLE6 in a non-simply-connected domain with finitely many boundary components, since

such a domain looks locally like a simply-connected domain.

This property implies the following "global" restriction properties. For convenience

only, we will state them under some assumptions on the boundaries of the domains.

COROLLARY 2.3 (splitting property). Let D denote a simply-connected domain such

that aD is a Jordan curve. Let a, band b' denote three distinct points on aD, and let

I denote the connected component of aD\{b,b'} that does not contain a. Let (Kt, t;;::O)

(resp. KD denote an SLE6 in D from a to b (resp. from a to b'). Let T (resp. T') denote

the first time at which K t (resp. I?:) intersects I. Then (Kt, t<T) and (K:, t<T') have

the same law up to time change.

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COROLLARY 2.4 (restriction property). Let D*CD denote two simply-connected

domains, and assume that aD is a Jordan curve. Suppose that I:=aD*\aD is connected.

Take two distinct points a and b in aDnaD*\l.

Let (Kt, t:;::O) denote SLE6 from a to b in D, and T:=sup{t:Kt nI=0}. Similarly,

let (K;, t:;::O) be SLE6 from a to b in D*, and T*:=sup{t:K;nI=0}. Then, (Kt, t<T)

and (K;, t<T*) have the same law up to time change.

In the present paper, we will use these results when D is a rectangle.

Proof of Corollary 2.3 (assuming Theorem 2.2). This is just a consequence of the

fact that in Theorem 2.2 with D=H and bounded N, one can replace 'Y by,8(s):='Y(l-s).

Then we get that the law of SLE6 in N from 0 to 'Y(O) is that of a time change of SLE6

in N from 0 to 'Y(1) up to their hitting times of 'Y. The result in a general domain follows

by mapping it conformally onto a nice neighborhood N with a mapped to 0 and {b, b'}

to b(O), 'Y(1)}. D

Proof of Corollary 2.4 (assuming Theorem 2.2). By approximation, it suffices to

consider the case where I is a simple path. Let f denote a conformal map from D

onto H, with f(a)=O and f(b)=oo. Define'Y in such a way that 'Y[O, l]=f(1); note that

D*=f-l(NC'Y», As bE8D*\I, NC'Y) is unbounded. Hence, by Theorem 2.2, the law of

SLE6 in D from a to b stopped when it hits I is (up to time change) the same as that of

SLE6 in D* from a to b stopped when its closure hits I. D

In order to prove Theorem 2.2, we will establish

LEMMA 2.5. Under the assumptions of Theorem 2.2, define for any fixed s<l,

Ls ='Y(O, s] and

T= sup{t:;:: 0: K t nLs = 0}.

For all t~T, let gs,t denote the conformal homeomorphism taking H\(KtULs) onto H

with the hydrodynamic normalization. Then, the process (9s,t, t<T) has the same law

as a time change of SLE6 in H\Ls starting at 9s,0-9s,0(0), up to the time when the

closure of its hull intersects Ls.

Proof of Theorem 2.2 (assuming Lemma 2.5). In the setting of the lemma, let

h () = 9s,t(Z)-9s,0(0) s,t Z , (0) .

9s ,0

By Lemma 2.5 and Proposition 2.1, tNhs,t has the same law as a time change of SLE6

starting at hs,o. Note that hs,o(O)=O, h~,o(O)=l. Hence, it follows easily that for all

ZENC'Y),

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By continuity, if we let h1,t=lims--+i hs,t, then tt--+h1,t has the same law as a time-changed

SLE6 (in N) started from 1/JN' The proof is completed by noting that the hull of hi,t

is K t . 0

The idea in the proof of Lemma 2.5 is to study how the process gs,t changes as s

increases. For this, we will need to use some of the properties of solutions to (2.1) where

WI< is replaced by other continuous functions, and to study how (deterministic) families

of conformal maps can be represented in this way with some driving function.

2.3. Deterministic expanding hulls

2.3.1. Definition and first properties. If (Ut , tE[O, a]) is a continuous real-valued function,

then the process defined by

(2.3)

and go(z)=z will be called the Lowner evolution with driving function Ut . Note that gt

satisfies the hydrodynamic normalization (2.2). Moreover,

(2.4)

for some functions aj(t), j=2,3, .... As above, we let DtcH denote the domain of gt,

and let Kt:=H\Dt . K t will be called the expanding hull of the process gt.

We now address the question of which processes K t can appear as the expanding

hull driven by a continuous function Ut . We say that a bounded set KcH is a hull if

H\K is open and simply-connected. The Riemann mapping theorem tells us that for

each hull K, there is a unique conformal homeomorphism gK: H\K -+H which satisfies

the hydrodynamic normalization (2.2). Let

A(K) =A(gK):= ~ lim Z(gK(Z)-Z); z--+oo

that is, g(z)=z+2A(g)Z-1+ ... near 00. Observe that A(K) is real, because 9K(X) is

real when xER and Ixl is sufficiently large. Moreover, A(K)?O, because Im(z-9K(z))

is a harmonic function which vanishes at infinity and has non-negative boundary values.

Note that

A(goh) = A(g)+A(h)

if 9 and h satisfy the hydrodynamic normalization. It follows that A( K) ~ A( L) when

KcL, since 9L=ggK(L\K)ogK.

The quantity A(g) is similar to capacity, and plays an analogous role for the equation

(2.1) as capacity plays for Lowner's equation.

868

246 G, F, LAWLER. O. SCHRAM/vI AND W, WERNER

THEOREM 2.6. Let (Kt , tE[O, aJ) be an increasing family of hulls. Then the follow

ing are equivalent:

(1) For all tE [0, aJ, A(Kd=t, and for each E>O there is a (5)0 s1J,ch that for each

tE[0,a-8] there is a bounded connected set SCH\Kt with diam(S)<E and such that S

disconnects Kt+J \ K t from infinity in H \ K t .

(2) There is some continuous U: [0, a]-tR such that K t is driven by Ut .

In [37] a similar theorem is proved for Lowner's differential equation in the disk.

Note that K t may change discontinuously, in the Hausdorff metric, as t increases.

For example, consider Kt:={exp(is):O<s~t} when t<7r, and R7f:={ZEH:lzl~l} and

K t+7f :=K1I'U(-1,-1+itj, t>O, say. and let Kt:=kp(t) where ¢> is chosen to satisfY

A(K.p(t))=t.

LEMMA 2.7. Let r>O and xoER, and suppose that K is a hull contained in the

disk {z: Iz-xol<r}. Then

I -1( ) 2A(K) I CrA(K)

gK z -z+ ~ 1 12 z-xo Z-Xo

for all zEH with Iz-xol>Cr, where C>O is an absolute constant.

Prool of Lemma 2.7. For notational simplicity, we assume that xo=O. Clearly, this

does not entail any loss of generality. By approximation, we may assume that K has

smooth boundary. Let Ie R be the smallest interval in R containing {gK(X): xEoKnH},

and let f:=g1/' Let II be the restriction of I to I. Let J* denote the extension of I to

C \ I, by Schwarz reflection. The Cauchy formula gives

27rij*(w) = 1 1*(z) dz+ [ JI(x)- Ji(X) dx, Izl=R z-w if x-w

provided that R>lwl, R>max{lxl:XEI} and WEC\!o Since J*(z)=z-2A(K)z-1+ ...

near 00,

lim 1 J*(z) dz = lim 1 _z_ dz = 27riw. R-+oo Izl=R z-w R-+oo Izl=R z-w

Consequently, we have

J*(w)-w= ~ [Im(JI(x)) dx. 7r if x-w

Multiplying by wand taking W-tOO gives

A(9K)=-A(j*)=2- [Im(JI(x))dx. 27r if

(2.5)

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Moreover,

f*(w)-w-2A(J*)w- 1 = ~ /, Im(/J(x)) (_I_+~) dx, 7r I X-W W

and therefore

1f*(w)-w-2A(f*)w-11 ~ ~ /, Im(/J(x)) sup{l(x-w)-l+ w-ll : xEI} dx 7r I

Hence, the proof will be complete once we demonstrate that there is some constant Co

such that IC[-cor,cor]. This is easily done, as follows. Define G(Z):=9K(rz)/r for

Izl>I, and write G(z)=z+alz-1+a2z-2+ .... The Area Theorem (see, e.g., [41]) gives

I;;:I:;:tilajI2. In particular, lajl~I for j;;:l. Consequently, we have IG(z)-zl~I for

Izl;;:2. By Rouche's theorem (e.g. [41]), it follows that G({lzl;;:2}):::){lzl>3}. Conse

quently, 9K(H\K):::){lzl>3r}, which gives IC[-3r,3r]. D

For convenience, we adopt the notation

Kt,u := 9Kt(Kt+u \Kt ).

Proof of Theorem 2.6. We start with (1) implies (2). Let R:=sup{lzl: zEKa} and

Q:={ZEH: Izl >R+2}. Let t, 6, c: and S be as in the statement of the theorem, and let

sEaS. Suppose that c:<1 and rE[C:, v'c]. Then there is an arc f3r of the circle of radius

r about s such that f3rcH\Kt and K t URuf3r separates Kt+5\Kt from Q. It therefore

follows that the extremal length of the set of arcs in H\Kt which separate Kt+5\Kt

from Q in H\Kt is at most const/log(l/c:). (For the definition and basic properties of

extremal length, see [1], [31]. The terms extremal length and extremal distance have the

same meaning.) Since extremal length is invariant under conformal maps, it follows that

the extremal length of the set of paths in H that separate K t ,8 from 9K,(Q) is at most

const/log(I/c:). Because the diameter of 9Kt (H\Q) is bounded by some function of R (this follows since 9K t has the hydrodynamic normalization), we conclude that at least one

of these arcs has length less than const/log(I/c:). Consequently, this is a bound on the

diameter of Kt,(j. Observe that this bound is uniform for all tE [0, a-6]. For each t<a, we

then define Ut to be the point in the intersection nu>o Kt,u' We have an upper bound on

diam(Kt,(j) which tends to zero uniformly as 6--+0, and therefore lim8--to9Kt,.(Z)-z=0

uniformly for ZEH\Kt ,8 and t~a-8. This implies that Ut is uniformly continuous on

[0, a) and can be extended continuously to [0, a]. Now let zoEH\Ka' Then there is some c>O such that Im(9Kt(zO» >c for all tE[O, al,

Lemma 2.7 applied with K =Kt,u, Z=9K t+,,(Zo) and xo=Ut+u gives

9K t(ZO)- 9Kt+u(Zo) + 2 --+ ° U 9K t+u(Zo)-Ut+u

870


as 8-+0. As gK,(ZO) and Ut are continuous in t, we may therefore conclude that

which gives (2).

The proof that (2) implies (1) is easy. Let £>0. Given O~t~t+u<a, let Q(t,u):=

u+max{U(t')-U(tl):t',t"E[t,t+u]}. Observe that diam(Kt,u)-+O if Q(t,u)-+O, and

Q( t, u) -+0 if u-+o. Consequently, the extremal length of the set of paths separating Kt,u

from {zEH:lzl>l} in H goes to zero as Q(t,u)-+O. This implies that there is a path

f3 in this set such that diam(gK~(,8»)<£, provided that u is small. We then just take

S=gK~(f3). 0

2.3.2. Time-modified expanding hulls and restriction. Let (Kt, tE [0, a]) denote a

family of hulls, and suppose that there is a monotone increasing homeomorphism ¢:

[0, aj-+[O, aj such that (Kq,(t),tE[O,a]) is an expanding hull driven by some function

tt---+fJt . If additionally ¢ is continuously differentiable in [0, aj and ¢'(t) >0 for each

tE [0, a], then we call (Kt, tE [0, aD a time-modified expanding hull, with driving function

Ut :=fJ4>-l(t). Note that, in this case, ¢-l(t)=A(Kt}, and that

!;) () _ 28t A(Kt ) Vt9K, Z - () TT' 9K, Z -Ut

(2.6)

Note that in our terminology, an expanding hull is always a time-modified expanding

hull.

LEMMA 2.8. Let (Kt, tEla, aD be a time-modified expanding hull, with driving func

tion (Ut , tE[O,a]). Let D be a relatively open subset of H which contains K a , and set

DR:=DnR. Let G: D-+H be conformal in D\DR and continuous in D, and suppose

that G(DR)cR. Then (G(Kt), tE [0, a]) is a time-modified expanding hull. Moreover,

(2.7)

Proof. We first prove (2.7). The proof will be based on (2.5). Note first that if

K'=aK then A(K')=a2A(K). Therefore, we may assume that G'(Uo)=l. Similarly,

with no loss of generality, we assume that Uo=G(Uo)=O. By the reflection principle,

G is analytic in D.

Set Kt:=G(Kt ). Let ItcR be the interval corresponding to aKtnH under gK" and

let it be the interval corresponding to aKtnH under gR. Let £>0, and let De:={ZED:

11-G' (z) 1 < e: }. Let f3 be some arc in De \ {O} that separates 0 from 00 in H. Consider

the map

871


It is well defined in a neighborhood of It provided that K t n {3 = 0 (for instance), and this

holds when t is small. This map may be continued analytically by reflecting in the real

axis, and therefore the maximum principle implies that

sup{h~(x): XEIt } ~sup{lh~(z)1 : zE9K,({3)}

when K t n{3=0. Note that 9Kt(Z)-Z-+O and 9R.(z)-z-+O as t~O, and therefore

9~ (z)-+l and g'- (z)-+l on {3. Consequently, for small t we have , K t

sup{h~(x): XEIt } < 1+2c. (2.8)

Note that for z close to Uo=O we have Im(G(z))~(l+c)Im(z). Using (2.5), this inequal

ity and (2.8), we get

= ~! Im(Go9K\X))h~(x) dx 21f It t

~ ~! (l+c) Im(gKl(x))(1+2c) dx 21f It t

= (1+c)(1+2c)A(Kt )

for small t>O. (Note that 9:K;(x) is not defined for every xEit, but it is defined for almost

every x E It.) By symmetry, we also have a similar inequality in the other direction. This

proves (2.7).

By Theorem 2.6, to show that G(Kt) is a time-modified expanding hull, it suffices

to show that A(G(Kt)) is continuously differentiable in t, with derivative bounded away

from O. Let Gt:=9RtoGo9:K~. Then G t is analytic in 9Kt(D\Kt ) and depends continu

ouslyon t. Hence G~(Ut) is continuous in t. Since A(G(Kt+u))=A(G(Kt))+A(Gt(Kt,u)),

it follows that atA(G(Kt))=G~(Ut)2atA(Kt), which completes the proof. 0

For future reference, we note that when 9t = 9K, satisfies the differential equation

(2.6), we have the formula

(2.9)

which is obtained by differentiating (2.6) with respect to z.

872

250 C. F. LAWLER. O. SCHRAMM AND W. WERNER

2.3.3. Pairs of time-modified expanding hulls. We now discuss the situation where

there are two disjoint expanding hulls.

Let (Ls, sE [0, soD and (Kt, tE [0, toD be a pair of time-modified expanding hulls such

that Lso nKto =0. Let gs.t:=gLsUK,) gt :=gK" gs:= gL s and a(s, t):=A(gs.d. Then for

each 8E [0, sol and tE [0, to] we have

Therefore,

where st--+U1(s, t) is the driving function for the time-modified expanding hulls st--+gt(Ls). Similarly,

where tt--+U2 (s, t) is the driving function for the time-modified expanding hulls tt--+ gs (Kt).

Although we do not know that g;1(U2 (O, t)) is well defined, g8.t o g;1 is analytic in a

neighborhood of U2 (0, t), by the reflection principle. Hence, it is clear that U2 (s, t)=

gs,tQg;1(U2(0, t)) (see, for example, the construction of Ut in the proof of Theorem 2.6),

and therefore

(2.10)

We will now prove the formula

fJ fJ ( )_ -48s a(s,t)8t a(s,t) s t a s,t - (U2(S,t)-U1(s,t))2' (2.11)

From (2.7) we have

and using (2.9), we obtain

8 () -48s A(gs) 8s log t a O,s = (gs(U2(O,O))-U1(S,O))2

-48s a(O,8) (U2(S, 0) - U1(8, 0))2'

This verifies (2.11) for the case t=O. The general case is similarly obtained.

873


2.4. Proof of Lemma 2.5

We will now prove Lemma 2.5; this is the core of the proof of the locality property.

We assume that T [O,sl]--+H is a continuous simple path with ')'(O)ER\{O} and Ls:= ')'(0, SI] CH. With no loss of generality, assume that')' is parameterized so that A(Ls)=s.

By Theorem 2.6, (Ls, sEra, SI]) is a time-modified expanding hull, and by Lemma 2.8,

for each s, tl---'tgL.(Kt ) is a time-modified expanding hull, and for each t, sl---'tgKt(Ls ) is

a time-modified expanding hull. Let tI---'tW(s,t) be the process driving tl---'tgL.(Kt), let

U(s, t) be the process driving sl---'tgK.(Ls), and let yt= w(a, t) be the process driving Kt .

As above, let a(s, t)=A(gs,d. For simplicity, W(s, t) will be abbreviated to W, U(s, t) to U, a(s, t) to a, etc.

Our aim is to show that (W(Sb t), t~a) is a continuous martingale (up to the stop

ping time T), and that its quadratic variation (for background on stochastic calculus,

see, e.g., [15], [39]) is

Indeed, if this is true, let ¢(t) be the inverse of the map tl---'ta(sb t) -a(sl' 0), and define

W(t)=W(Sl,¢(t)). Then W(-~t) is a Brownian motion, so that tl---'tgs1 ,o(K4>(t))-gSl,O(0)

is an SLE6-process, as required. Note that this will in fact give a precise expression for

the time change in Lemma 2.5 and Theorem 2.2.

Before giving the mathematically rigorous proof, we first present a formal, non

rigorous derivation of the fact that W( s, . ) is a martingale. In this derivation, r;, will be

kept as a variable, in order to stress where the assumption r;,=6 plays a role (it will not

be so apparent in our proof).

Non-rigorous argument. The first goal is to show that the quadratic variation (W)t of tl---'t W(s, t) satisfies

for each 8, t. It is clear that this holds when 8=0, since K t is SLE6. We have

Consequently,

Os Ot(W)t = OtOs (W)t

=2ot(osW, W)t

= 2ot(2(osa)(W - U)-l, Wh (by (2.10))

= -4(osa)(W -U)-2ot (Wh (by Ito's formula)

= (Ota)-l(osota)Ot{W)t (by (2.11)).

(2.12)

874

252 G.F. LAWLER, O. SCHRAMM AND W. WERNER

which means that Ot(W)t!ota does not depend on s. Since (2.12) holds when s=O, this

proves (2.12).

We now show that tl---+ W( s, t) is a martingale. The dt-term in Ito's formula for the

ot-derivative of

is 2otosa asa asa W-U +2 (W-U)3 at (W)t- 4 (W-U)3 ata,

where the first summand comes from differentiating osa, the second summand is the

diffusion term in Ito's formula, and the last summand comes from differentiating with

respect to U and using (2.10) for atu. Using (2.11) and (2.12), this becomes

(3- 1;;;) OtOsa 2 W-U'

which vanishes when ,,;=6. Hence t 1---+ Os W(s, t) is a martingale. As

it follows that tl---+ W(s, t) is a martingale. This completes the informal proof.

The problem with the above argument is that we do not know that tl---+ W( s, t) is a

semi-martingale, and hence cannot apply stochastic calculus to it. Moreover, we need to

check that there is sufficient regularity to justify the equality as at (W)t = at as (Wk

To rectify the situation, set

t'

Yes, t'):= W(s, 0)+ 10 ylata(s, t) dyt.

Then tl---+V(s, t) is clearly a martingale. The rest of this subsection will be devoted to

the proof of the fact that V = W. Recall that

We will need the following fact:

LEMMA 2.9. There exists a continuous version of Von [0, sll x [0, T).

Proof. In order to keep some quantities bounded, we have to stop the processes

slightly before T. Let us fix c:E (0,1), and define

T{ = in£{ t > 0: inf IW(s, t)- U(s, t)l:( E}. S(:SI

875


Define for any s ~ S1 and to ~ 0

~ ~ rmin(to,Tf) V(s, to) = vc(s, to):= Jo ~ dyt.

As T 1/ n sup 1 =T,

n~1

it is sufficient to show existence of a continuous version (on [0, sd xR+) of V. Let

a = a(s, t) := l{t~Tn ~.

Note that Ota(O, t)=osa(s, 0)=1. Hence, from (2.11) it follows that osa~l and ota~l

for all S~Sl, t<T. Using (2.11) again, we get

Hence, for all t~O, for all s, s' in [0, S1],

la(s, t) -a(s', t)1 ~ 2c-2 Is-s'l.

But

and using, for instance, the Burkholder-Davis-Gundy inequality for p=4 (see, e.g.,

[39, IVA]), we see that there exists a constant Cl=C1(c) such that for all to, t~~O and

s, s' E [0, S1],

E[(V(s, to)-Vel, t~))41 ~ cIE[(to-t~fl +CIE[ (foto(s-s'? dt J] ~ Cl(to _t~)2 +C1t~(S-S')4,

and the existence of a continuous version of V then easily follows from Kolmogorov's

lemma (see, e.g., [39, 1.(1.8)]). 0

From now on, we will use a version of V that is continuous on [0, S1] x [0, T). Define

T~ := inf {t ~ 0: sup IV(s, t) - W(s, t)1 ~ I}, S~Sl

T3:=inf{t~O: inf IV(s,t)-U(s,t)l~c}, S~Sl

TC:= min(Tf, T~, T3)'

876

254 G. F. LAWLER. O. SCHRAMM AND W. WERNER

Note that for all S~Sl and t<T,

8 !f'G _ -2va;G, osa sYUta - (W-U)2 .

The process osVota remains bounded before T€ (uniformly in S~Sl)' and it is a measur

able function of (s, t). By Fubini's theorem for stochastic integrals ([15, Lemma III.4.1]),

we have that for all SO~Sl' for all to;;;::O, almost surely

lSi: -~~~;a dYt ds = lt~(ls08sy1i;;; dS) dYt

t'

= 10 0 (JOta(so, t) - JOta{O, t) ) dYt

= V(so. t~)- yeO, t~) - (W(so, 0)-W{O, 0)),

where to:=min(to, T€). On the other hand, using Ito's formula, we now compute

2osa(s, to) _ 2 lt~ 2osava;a d},; U{s,to)-V(s,to) - U(s,O)-V(s,O) + 0 (U-V)2 t

(2.13)

21t~(OtOsa _ osaot U OsaOt(Vh) dt (2.14) + 0 U - V (U - V)2 + (U - V)3

t' t'

=-8sW(s,0)+ 10 °b1dYt+ 1 o(V-W)b2dt,

where (using Ot (V}t=Otaot (Y}t=",ota=6{)ta)

- 2osavota bl(s. t):= (U _ V)2 '

48sa8t a ( W - V) b2(s,t):=(U_W)2(U_V)2 5+3 V _ U .

Note that for all S~Sl and t~T€,

Ib2 (s, t)1 ~ 16e-5 •

By integrating (2.14) with respect to s and subtracting (2.13) from it, we get

( ') ( ') 1s0 28s a(s, to) V so,to -V O,to + (') V( ') ds o U s, to - s, to

18 0 t~ = 0 io (V - W)b2 (s, t) dt ds

rO(lt~ -2va;G, {) alto - ) + io 0 (W-U); dYt+ 0 b1(s,t)dYt ds

(2.15)

877


where (after some simplifications)

2osaJota bl(s, i):= (V -U)2(W -Up (U - W)+(U - V)).

Note that for all sE [0, Sl] and i~TE:,

But we know on the other hand that

( ') (') l s0 2osa(s,io) W so,to -W O,to = We ')-U( ') ds. o s,io s,to

(2.16)

Subtracting this equation from (2.15), one gets

to V(so, t~) - W(so, t~) = 10 b3(s, t~)(V(s, i~) - W(s, t~)) ds

So t~ {SO t~

+ 110 (V - W)b2 dtds+ 10 10 (V - W)b 1 dyt ds,

where -2osa

b3 (s, t):= (U _ W)(U - V)'

Again b3 remains uniformly bounded before TE:.

We now define

H(s, t) = V(s, t)-W(s, t).

Hence, for all to~T,

H(so, to) = lsob3(S, to)H(s, to) ds+ lSj:b2 H dsdt+ l

Sj:bl H dsdyt

and Ibl l,lb21,lb31 are all bounded by some constant c2=c2(c) on [O,sl]x[O,TE:]. This

equation and an argument similar to Gronwall's lemma will show that H = O.

Let us fix tl>O. For any t;::O, define

ret) = min(t, tl, TE:).

We will use the notation ro=r(to). It is easy to see that there exists a c3=c3(c, h, sd such that for all to;::O and soE[O, Sl],

878

256 c. F. LAWLER, O. SCHRAMM AND W. WERNER

The Burkholder-Davis-Gundy inequality for p=2 (see, e.g., [39, IVA]) shows that there

exist constants C4=C4(C, tl, 81) and C5=C5(C, h, 81) such that for all sE [0, SI] and to ~O,

roro ~ C5 io io E[H( 8, T{t»2] ds dt.

Let us now define

h(so, to) = E [ sup H(so, T(t»2]. t";;to

Then

h(so, to) ~ C6 (fosoh(S, to) ds + fothsoh(s, t) ds dt)'

We also know that h(s, t) is bounded by 1 (because IHI ~ 1). Hence, it is straightforward

to prove by induction that for all soE[D, SIJ, to~D and p=l, 2, ... ,

h( ) c~sl:(1+to)P So, to ~ I ' p.

so that h(so, to)=D. In particular (using the continuity of V and W), this shows that

W=V almost surely on all sets [O,sIJx[O,min(tl,TE)]. As this is true for all € and tb

we conclude that V = W on [0, Sl] x [D, T). Lemma 2.5 follows, and thereby also Theo

rem 2.2. 0

3. Exponents for SLE6

3.1. Statement

In the present section, we are going to compute intersection exponents associated with

SLE6 ·

Suppose that De C is a Jordan domain; that is, aD is a simple closed curve in C.

Let a, bEaD be two distinct points on the boundary of D. As explained in §2.1, the

SLE6 (Kt , t~D) from a to b in D is well defined, up to a linear time change.

Now suppose that leaD is an arc with bEl but a¢l. Let

T1:= sup{t ~ 0 :Ktnl = 0}.

By Corollary 2.3, up to a time change, the law of the process (Kt, t < T1) does not change

if we replace b by another point b'EI. Set

S=S(a,I,D):= U K t , t<7"1

879


and call this set the hull from a to I in D. It does not depend on b.

Suppose that L>O, and let R=R(L) denote the rectangle with corners

(3.1 )

Let S denote the closure of the hull from A4 to [AI, A2J U [A2' A3J in R. In the following, we will use the terminology 7r-extremal distance instead of "7r times

the extremal distance". For instance, the 7r-extremal distance between the vertical sides

of R in R is L.

When Sn [AI' A 2 ] =0, let £ be the 7r-extremal distance between [AI, A4] and [A2' A3J in R\S. Otherwise, put C=oo.

In the sequel, we will use the function

u(>.) = i (6)'+1+J24>'+1). (3.2)

The main goal of this section is to prove the following result.

THEOREM 3.1.

E[l{L<=}exp(->,C)] =exp(-u(>.)L+O(l)(>.+l)) as L-+oo, (3.3)

for any >.~O (where 0(1) denotes an arbitrary quantity whose absolute value is bounded

by a constant which does not depend on L or .A).

In particular, when >.=0,

p[Sn [AI, A2J = 0] = P[C < 00] = exp( -!L+O(l)) as L -+ 00. (3.4)

3.2. Generalized Cardy's formula

By conformal invariance, we may work in the half-plane H. Map the rectangle R con

formally onto H so that Al is mapped to 1, A2 is mapped to 00, A3 is mapped to 0,

and then the image x=x(L)E(O, 1) of A4 is determined for us. Let K t be the hull of an

SLE6-process gt=gKt in H, with driving process W(t) , which is started at W(O)=x (that

is, K t is a translation by x of the standard SLE6 starting at 0). In order to emphasize the

dependence on x, we will use the notation Px and Ex for probability and expectation.

Set

To:= sup{t ~ 0: Ktn( -00,0] = 0},

TI := sup{t ~ 0: K t n[l, (0) = 0},

T:= min{To, Td.

880


As will be demonstrated, To, Tl < 00 a.s. Let

for t<T, which is just gt renormalized to fix 0, 1 and 00. It turns out that fT:= limt/,T ft

exists a.s. On the event Tl > To, fT uniformizes the quadrilateral

(H\KT; 1, oo,min(KTnR),max(KTnR))

to the form

(H; 1, 00, O,h(max(KTnR))).

Therefore, we want to know the distribution of

1-h(max(KTnR)),

and how it depends on x (especially when XE(O, 1) is close to 1). In this subsection,

we will calculate something very closely related: the distribution of f~(I) and how it

depends on x.

Set

for b~O and XE(O, 1). Recall the definition of the hypergeometric function 2Fl (see, e.g.,

[30]):

F ( ) ~ (aO)n(al)n n 2 I aO,al,a2jX = ~ ( ) I X ,

n=O a2 nn.

where (a)n=I1;=l(a+j-l) and (a)o=l. Note that 2Fl(ao,al,a2jO)=1.

THEOREM 3.2. For all b~O, xE(O, 1),

-2b 5 ' ..fii2 r(-+b) / b" "" , A(x b) = 6 A xl 6+ F ('!+b .!+b 1+2b' x)

, r(~)r(l+b) 2 1 6 ' 2' "

where

and 2Fl is the hypergeometric function.

Setting b=O, we obtain Cardy's formula [7]. Thus, this result can be thought of as

a generalization of Cardy's formula.

Note that Theorem 3.2 determines completely the law of 1{To<T1}fHl). In par

ticular, the Laplace transform of the conditional law of log Ijf~(l) given {To<Tt} is

A(I-x, b)j A(I-x, 0).

881


Proof. We first observe that T1<00 almost surely. Indeed, gt(l)-W(t) is a Bessel

process with index i, with time linearly scaled, and hence hits 0 almost surely in finite

time (e.g. [39]). Similarly, gt(O)- Wet) hits 0 almost surely. It is clear that Tl is the time

t when gt(1)-W(t) hits 0, and To is the time when gt(O)-W(t) hits O. It follows from

Theorem 2.6 and XE(O, 1) that almost surely To#Tl' We may also conclude that

The next goal is to prove that

lim f:(l) > 0 if and only if To < T1• t/,T

(3.5)

(3.6)

If To<TI, then KT n[1,00]=0. Therefore h is defined and conformal near 1, and

f~(l) >0, by the reflection principle. On the other hand, if Tl <To, then KTn [1, (0)#°. We claim that

1 fj. KTl almost surely; (3.7)

since KTl n [1, (0)#°, this means that KTl separates 1 from 00 in H. Indeed, let ¢>: H-+H be the anti-conformal automorphism that fixes x and exchanges 1 and 00. Tl < 00 a.s.

and SUPt<T1IW(t)I<00 a.s. imply that KTl is bounded a.s., which is the same as saying

that ¢>(Kt ) stays bounded away from 1 as t/,T1 . But Corollary 2.3 and invariance under

reflection imply that up to time Tl the law of K t is the same as a time change of the law

of ¢>(Kt). Hence, a.s. K t stays bounded away from 1 as t/,T1, proving (3.7). It follows

from (3.7) that limVT1f:(1)=0 a.s. on the event Tl <To (observe that, given (3.7), the

extremal length from a neighborhood of 0 to a neighborhood of 1 in H\Kt tends to 00

as t/,T1 ), and (3.6) is established.

Define the renormalized version of W (t),

and the new time parameter

Z(t).- W(t)-gt(O) .- 9t(1)-9t(0) '

Set sO:=limVT set). Since Tl #To a.s., inf{9t(1)-9t(0): t<T} >0 a.s., and hence 80<00

a.s. Let t(s) denote the inverse to the map tt-+8(t). A direct calculation gives

882


and

We now use the notation

Z(s):= Z(t(S)), 18(z):= ft(s)(z),

Then,

dZ8=dX8+ 2(1-2Z(s))ds =dX8+(~- 2 )dS, (3.8) Z(s)(l-Z(s)) Z(s) 1-Z(s)

where (Xs, s~O) has the same law as (W(t), t~O); Le., it is a Brownian motion with time

rescaled by a factor of 6. Also,

(3.9)

These two equations describe the evolution of 18(z), Note that s(T)=so is the first time

at which Z(s) hits 0 or l.

We now assume that b>O. Differentiating (3.9) with respect to z gives

- -2 2 2 Os (log j;(z)) = (Z(s)-ls(z))2 - 2(s) - 1-Z(s) (3.10)

(the Cauchy integral formula, for example, shows that we may indeed differentiate, but

this is also legitimate since Os and Oz commute in this case). We are particularly inter

ested in

which satisfies -2 2 2

osa(s) = (Z(s)-l)2 - Z(s) - 1-Z(s) (3.11)

Note that the equations (3.8) and (3.11) describe the evolution of the Markov process

(Z(s), a(s)). The process stops at So. Define

y(x, v) := E[exp(ba(so)) I 2(0) = x, a(O) = v],

where the expectation corresponds to the Markov process started from Z(O)=x and

a(O)=v. From the definition of y it follows that

y(x, 0) = A(l-x, b),

883


since limvT it = h in a neighborhood of 1 on the event To < Tl and (3_6) holds. It is

standard that such a function y( x, v) is Coo, and the strong Markov property ensures

that the process

Y =y(Z(8), 0'(8))

is a local martingale. The drift term in Ito's formula for dY must vanish, which gives

As

we get

y(x, v) = exp(bv)y(x, 0).

Set

h(x):= y(l-x,O) =A(x, b),

so that y(x,v)=exp(bv)h(l-x). Hence (3.12) becomes

-2bh(x)+2x(1-2x}h'(x)+3x2 (1-x)h"(x) = 0.

The second statement in (3.5) implies that

while

lim hex) =0, x '\.0

lim hex) = 1 x/,1

(3.12)

(3.13)

holds, since when x is close to 0, KTo is likely to be small, by scale invariance, for example.

The differential equation (3.13) can be solved explicitly by looking for solutions of the

type h(x)=xCz(x): two linearly independent solutions are (i=l, 2)

where

Recall that 2Fl(aO, aI, a2;0)=1. The function h(x) must be a linear combination of hI

and h2. However, lim x'\. 0 h(x) =0= lim x'\.O h1(x), but lim x '\.0 h2 (x)=oo. Hence, h(x}=

ch1(x) for some constant c. The equality h(I)=l and knowledge of the value at x=l of

884

262 C. F. LAWLER, O. SCHRAMM AND W WERNER

hypergeometric functions (see, e.g., [30]) allows the determination of c, and establishes

the theorem in the case b>O. The case b=O follows by taking a limit as b'\tO. D

Remark. With the same proof, Theorem 3.2 generalizes to SLEK with 11:>4, and

gives

where b '= J(x;-4)2+16I1:b K' 2x; ,

C(b x;):= r(~-6/x;+bl<)r(~+2!x;+bl<). , r(1-4/1I:)r(1+2bK )

3.3. Determination of the SLEa-exponents

For every t ~ 0, set

M t := max(Kt nR).

The following lemma shows that our understanding of the derivative fHl) gives infor

mation on Jr(MT ) itself.

LEMMA 3.3. In the above setting, let NT:=Jr(MT)' For b~O, set

Then

(3.14)

Note that 8(x,b) is close to the quantity we are after, since -log(I-NT) is approx

imately the extremal length of the quadrilateral

(H\KTjmin(KTnR), max(KTnR), 1,00).

Proof. It follows easily from (3.10) that ff(z) is non-decreasing in z (viewed as a

real variable), as long as z ~ M t . Therefore,

I-NT = (l fHz) dz ~ (I-MT)fr(l) ~XfT(l). iMT

This gives the right-hand inequality in (3.14). To get the other inequality, consider some fixed x* > 1 (x* should be thought of as

close to 1; we will eventually take x*=I/(I-x)). Let x*=Jr(x*). Then

885


This gives

(3.15)

A simple scaling argument will give an upper bound of the left-hand side of this inequality

in terms of e. Let

T* = inf{t ~ 0: KtnR\(O, x*) tf0}.

Note that T~T*<oo a.s. and that T=T* if To<Tl. For each t~T*, let ft* be the

conformal map from the unbounded component of H\Kt to H, which fixes the points

00,0, x*. For all t~T, x*

ft(z) = h(x*) h(z).

Note that

h(x*) ~ x* = ft* (x*).

Then,

Hence,

(3.16)

since on the event To<Tt, we have T=T* and MT=MT*. However, by scale invariance,

when W(O)=l-x the random variable

has the same law as the random variable

does when W(O)=(l-x)jx*. Thus, combining (3.16) and (3.15) gives

(x*)b8{1-{1-x)jx*,b) ~ (x*-l)bA(x,b).

We take x*=lj(l-x), say, and get

8(2x, b) ~ 8{2x-x2 , b) ~ xb A{x, b),

which gives the left-hand side of (3.14). o

Proof of Theorem 3.1. We are now ready to derive Theorem 3.1 by combining The

orem 3.2 and Lemma 3.3.

886


In the setting of the theorem, let ¢>: R( L) -+ H be the conformal homeomorphism

satisfying ¢>(Al)=l, ¢>(A2 )=00 and ¢>(A3)=0. Set x=x(L):=¢>(A4)' By conformal in

variance, the law of £ is the same as that of the 7r-extremal distance 12* from (-00,0]

to (NT, 1) in H (with the notations of Lemma 3.3). Considering the map zr-+log(z-l)

makes it clear that

L = -log(l-x)+O(l), (3.17)

for L> 1, and similarly

C = -log(l-NT)+O(l). (3.18)

For L>l (note also that 12?;L),

E[l{£<oo} exp( -).12)] = E[l{£*<oc} exp( - )'C)]

=exp(O(l))E[l{NT<l}(l-NT)A] (by (3.18))

=exp(O(1))8(1-x, ).)

=exp(O().+l))(l-x)u(A) (by Lemma 3.3 and Theorem 3.2)

=exp(O().+ 1)) exp( -u().)L) (by (3.17)),

which completes the proof of Theorem 3.1. D

4. The Brownian half-plane exponents

We are now ready to combine the results collected so far and a "universality" idea similar

to that developed in [29] to compute the exact value of some Brownian intersection

exponents in the half-plane.

4.1. Definitions and background

In this short subsection, we quickly review some results on intersection exponents between

independent planar Brownian motions. For details and complete proofs of these results,

see [28], [29] . Suppose that n+p independent planar Brownian motions [31, ... , [3n and ')'1, ... , ')'P are

started from points [31(0)= ... =[3n(0)=0 and ')'l(O)= ... =')'P(O)=l in the complex plane,

and consider the probability fn,p(t) that for all j~n and l~p, the paths of [3j up to time

t and of ')'Z up to time t do not intersect; more precisely,

fn.p(t) := P [ C01 [3j [0, t]) n (Ql ')'z [0, t]) = 0].

887


It is easy to see that as t--+oo this probability decays roughly like a power of t. The

(n, p )-intersection exponent t;( n, p) is defined as twice this power, i.e.,

in,p(t) = (Jtr~(n,pl+o(l), t --+ 00.

We call t;(n,p) the intersection exponent between one packet of n Brownian motions

and one packet of p Brownian motions (for a list of references on Brownian intersection

exponents, see [28]). Note that the exponent ( described in the introduction is ~t;(1, 1). It turns out to be more convenient to use this definition as a power of v't, i.e., of the space

parameter. A Brownian motion travels very roughly to distance v't in time t: recall that

if (J is a planar Brownian motion started from 0, say, and TR denotes its hitting time of

the circle of radius R about 0, then for all 8>0, the probability that TR ¢,(R2- 8, R2+1i)

decays as R--+oo faster than any negative power of R. This facilitates an easy conversion

between the time-based definition of intersection exponents and a definition where the

particles die when they exit a large ball.

Similarly, one can define corresponding probabilities for intersection exponents in a

half-plane,

- . I . I in,p(t) := P[Vj ~ n, Vl ~ p, {JJ [0, t] nl' [0, t] = 0 and (JJ [0, t] UI' [0, t] C HJ,

where H is some half-plane containing the two starting points (in,p(t) will depend on H). In plain words, we are looking at the probability that all Brownian motions stay in the

half-plane and that all {J's avoid alll"s. It is also easy to see that there exists a ~(n,p) (which does not depend on H) such that

Note that ( described in the introduction is ~~(1) 1).

One can also define intersection exponents t;(nl, ... , np) and ~(nl' ... , np) involving

more packets of Brownian motions. (For a more detailed discussion of this, see [28]). For

instance, if BI, B2, B 3, B4 denote four Brownian motions started from different points,

the exponent ~(2, 1, 1) is defined by

P[the three sets B 1 [0,t]UB 2 [0,t], B 3 [O,tJ, B 4[0,t] are disjoint]

= C~(2,1,1)/2+o(1), t --+ 00.

One of the results of [28] is that there is a natural and rigorous way to generalize the def

inition of intersection exponents between packets of Brownian motions to the case where

each packet of Brownian motions is the union of a "non-integer number" of paths; for

888


the half-plane exponents, one can define the exponents ~(Ul' ... ,up), where Ul, ... ,up;:;:O.

These generalized exponents satisfy the so-called cascade relations (see [28]): for any

l::::;;q::::;;p-l,

(4.1)

Moreover, ~ is invariant under a permutation of its arguments.

There exists (see [28J, [29]) a characterization of these exponents in terms of the

so-called Brownian excursions that turns out to be useful. For any bounded simply

connected open domain D, there exists a Brownian excursion measure /LD in D. This is

an infinite measure on paths (B(t), t::::;;r) in D such that B(O,r)CD and B(O), B(r)EoD (these can be viewed as prime ends if necessary). xs:=B(O) and xe:=B(r) are the

starting point and terminal point of the excursion. One possible definition of /LD is

the following: Suppose first that D is the unit disc. For any 8>0 define the measure

ps on Brownian paths (modulo continuous increasing time change) started uniformly

on the circle of radius exp( -8), and killed when they exit D. Note for any 80> s, the

killed Brownian path defined under the probability measure ps has a probability 8 j 80 to

intersect the circle of radius exp( -so). Then, define

/LD:= lim (27fjs)ps. 8\,.0

One can then easily check that for any Mobius transformation ¢ from D onto D,

¢(/LD)=/LD. This makes it possible to extend the definition of /LD to any simply

connected domain D, by conformal invariance. These Brownian excursions also have

a "restriction" property [29J, as the Brownian paths only feel the boundary of D when

they hit it (and get killed).

Suppose for a moment that R = R( L ) C C is the rectangle with corners given by (3.1),

and that B is the trace of the Brownian excursion (B(t), t::::;;r) in R. Define the event

i.e., B crosses the rectangle from the left to the right. (Although /Ln. is an infinite

measure, /LR(E1 ) is finite.) When El holds, let R"1 be the component of R\B above B,

and let RB be the component of R\B below B. Let £B (resp. £"1) denote the 7f-extremal

distance between [Al,xsJ and [A2 ,xe J in RB (resp. [Xs,A4J and [xe ,A3 ]) in R"1. Then, for any a;:;:O and 0/;:;:0, the exponent ~(a, 1, a')=~(1, ~(a, a'» is characterized

by

EJL~ [IEl exp( -a£"1-a'£B)J = exp( -~(a', 1, a)L+o(L», (4.2)

when L-+oo, where EJL~ denotes expectation (that is, integration) with respect to the

measure /Ln.. Similarly,

(4.3)

889


See [29]. It will also be important later that € is continuous in its arguments, and that

"\t-+€(1,"\) is strictly monotone.

4.2. Statement and proof

For any P?O, we put

Vp = ~p(p+1).

Let V denote the set of numbers {vp:PEN}. Note that the smallest values in V are

0, ~, 1,2, 13° ,5,7.

We are now ready to prove the following result:

THEOREM 4.1. For any k?2, 0'1, ... , Uk-l in V, and for all UkER+,

It is immediate to verify that this theorem implies Theorem 1.1.

Remark. In [26], Theorem 4.1 is extended to all non-negative reals 0'1, ... , Uk.

Theorem 4.1 is a consequence of the cascade relations and the following lemma,

which is the special case of the theorem with k=2, 0'1 = i: LEMMA 4.2. For any "\>0,

€(i,.,\) = u("\) ,

where u("\) is given by (3.2).

Proof of Theorem 4.1 (assuming Lemma 4.2). Define U(A)=V24.,\+1-1, for all

A?O. Lemma 4.2 implies immediately that for all .,\?O,

u(€(i,"\)) = U("\)+2 = U("\)+U(i)

and (for all integer p) VP+l=€(i,VP)' The cascade relations then imply that for all in

tegers PI, ···,Pk-l,

This is (4.4). o Proof of Lemma 4.2. For convenience, we again work in a rectangle rather than in

the upper half-plane. Let R=R(L), and let S denote the closure of the hull of SLE6 from

A4 to [AI, A2J U [A2' A3J in R, as in Theorem 3.1. Let B denote the trace of a Brownian

890

268 C. F. LAWLER, O. SCHRAMM AND W. WERNER

excursion in R; we will call its starting point Xs and its terminal point Xe. Consider the

events

El = {xs E [AI, A4J and Xe E [A2' A3]},

E2 = {Sn[Al,A2J = 0},

E3 = E 1nE2n{SnB= 0}.

When E2 holds, let ,cs denote the 7r-extremal distance between the vertical edges of R

in R\S (that is, in the quadrilateral "below" S). Otherwise, let £s=oo.

When E1 holds, let R~ be the component of R\B above B, and let RB be the

component of R\B below B. Let 'cB (resp. ,c~) denote the 7r-extremal distance between

the vertical edges of R in R B (resp. in R ~), as before. When E3 holds, let 'cSB denote the

7r-extremal distance between the vertical edges of R in R'1 \S (that is, in the quadrilateral

"below S and above B").

Let A>O. We are interested in the asymptotic behavior of

f(L) = E[lE3 exp( -A'cSB)J

when L--+oo. By first taking expectations with respect to B (with the measure J.lR), and

using the restriction property (Corollary 2.4) for the domains R and R~, it follows that

as L--+oo,

f(L) = EB[Es[exp( -A'cSB)]]

= EB[exp( -u('>')'c~+O(l))l (by Theorem 3.1 and restriction to R~)

= exp( -~(1, u(.>.))L+o(L)) (by (4.3)).

On the other hand, we may first take expectation with respect to S. Given S, the law

of 'cSB is the same as that of 'cB' by complete conformal invariance of the excursion

measure (which is the analogue of the restriction property to the excursion measure,

see [28]). Hence, as L--+oo,

f(L) = Es[EB[lE3 exp( -A£SB)]]

= Es[EB[lE3 exp( -A£B)]]

= EB [Es[lE3 exp( -'>''cB)]J

= E B [Ps [E3 1 £~] exp( -A£'8)J

= EB [exp( -i£~ +0(1)) exp( -A£B) ] (by (3.4))

=exp(-~(i, 1,.>.)L+o(L)) (by (4.2))

= exp( -~(1,~(i, >'))L+o(L)),

891


by the cascade relations (4.1). Comparing with (4.5) gives ~(1,~U,A))=~(1,U(A)). Finally,

~(~, A) = u('\)

follows, since ,\1t--t~(1, A') is strictly increasing. o

5. Crossing exponents for critical percolation

It has been conjectured [42] that SLE6 corresponds to the scaling limit of critical perco

lation clusters. As additional support for this conjecture, we now show that it implies

the conjectured formula for the exponents corresponding to the probability that a long

rectangle is crossed by p disjoint paths or clusters of critical percolation ([13]' [8], [3]).

Let us first explain the conjectured relation between SLE6 and critical percolation.

Let DeC be a domain whose boundary aDeC is a simple closed curve. Let a,bEaD

be distinct points. Let 1'1 be the counterclockwise arc on aD from a to b, and let 1'2 be

the clockwise arc on aD from a to b. Let 8>0, and consider a fine hexagonal grid H in

the plane with mesh 8 j that is, each face of the grid is a regular hexagon with edges of

length 15, and each vertex has degree 3. For simplicity, assume that aD does not pass

through a vertex of H, and that a and b do not lie on edges of H. Color each hexagon

of H independently, black or white, with probability ~. Then the union of the black

hexagons forms one of the standard models for critical percolation (see Grimmett [14]

for percolation background and references).

Given the random coloring, there is a unique path f3eD that starts at a and ends

at b, such that whenever f3 is not on 1'1 it has a black hexagon on its "right", and whenever

f3 is not on 1'2 it has a white hexagon on its "left". This path is the boundary between

the union of the white clusters in D touching 1'2 and the black clusters in D touching 1'1.

Let f: D--+H be a conformal homeomorphism such that f(a)=O and f(b)=oo, and para

meterize f3 in such a way that A(j(f3[O, t]))=t. Let Dt be the component of D\f3[O, t] that has b on its boundary, and let Kt=D\Dt . The conjecture from [42] (stated a bit

differently) is that as 8--+0 the process (Kt, t~O) converges to SLE6 from a to bin D. In

light of this conjecture, the Locality Theorem 2.2 and its corollaries are very natural.

N ow consider an arc I e aD, which contains b but not a. Let b1 and b2 be the

endpoints of I, labeled in such a way that the triplet a, b1 , b2 is in counterclockwise

order around D. Let 1'{ e 1'1 be the counterclockwise arc from a to b1 , and let 1'~ e 1'2

be the clockwise arc from a to b2 • Let T be the first time such that f3(t)EI, and set

S:=Ut<TKt. Then the component G1 of aSnD joining 1'{ to I is a crossing in B from

1'{ to I which is "maximal", in the sense that any other crossing G e B from 1'{ to I is

separated by G1 from b2 in D.

892

270 C.F. LAWLER, O. SCHRAMM AND W. WERNER

Let L be large, and recall the definition of the rectangle 'R='R(L) with corners given

by (3.1). Let pEN+ and 0"=(0"1, ... ,lTp)E{black, white}p. Consider the event Cu('R) that

there are paths aI, ... , ap from [A4, AI] to [A2 , A3J in 'R such that each aj is contained

in the union of the hexagons of color O"j, there is no hexagon which intersects more than

one of these paths, and aj+l separates aj from [A},A2J in 'R whenj=1,2, ... ,p-1.

Take a1 to be the topmost crossing with color lTl, if such exists, let a2 be the topmost

crossing with color 0"2 which is below all the hexagons meeting aI, etc. Then Cu('R)

holds if and only if these specific aI, ... , a p exist. Note that after we condition on aI,

the hexagons "below" it are still independent and are black or white with probability ~.

Hence the following formula holds:

where 0"+1=(0"2,lT3, ... ,lTp), 'Rul is the union of the hexagons below aI, and CU+l('RuJ is

the event that there are multiple crossings with colors specified by 0"+1 from [A4, AI] to

[A2, A3J in nUl'

It is clear that P[C,,('R)J does not depend on the choice of the sequence lT, but only

its length. Moreover, the conjectured conformal invariance (or the conjecture that SLE6

is the scaling limit) implies that limo--+oP[Cu(D)] depends on the quadrilateral V only

through its conformal modulus. Hence define

fp(L):= lim P[C,,('R(L))l, O"E {black, white}p. 6--+0

We also set fp(oo):=O and fo(L):=l{L<CXl}' Let S be the SLE6-hull from A4 to I:=[A l ,A2lU[A2 ,A3l in 'R='R(L), as defined

in §3.1. Let 'R_ be the component of 'R\S which has Alan its boundary, and let

£ denote the IT-extremal length from [A4, All to [A2, A3l in 'R_. Note that £=00 if

Sn[A1 ,A2]i:0. Then we have

To completely justify this step requires more work, which we omit, since this whole

discussion depends on a conjecture anyway. The slight difficulty has to do with the fact

that having a crossing of a closed rectangle is a closed condition, and the probability of a

closed event can go up when taking a weak limit of measures. One simple way to deal with

this is to note that when the continuous process has a crossing in the rectangle 'R(L+c), every sufficiently close discrete approximation of it has a crossing of the rectangle 'R(L).

Consequently, induction and Theorem 3.1 give

fp(L) = exp( -(L+O(l))vp), L-7oo, (5.1)

893


where

(5.2)

as before. Here, the constant implicit in the O(l)-notation may depend on p.

Note also that if

(7 = (white, black, white, black, ... , white) E {black, white}2k-l,

then (in the discrete setting) the event C<T(n) is identical to the event that the rectangle

n is crossed from left to right by k disjoint white clusters.

The exponents (5.2) are those predicted in [8], [3].

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895


GREGORY F. LAWLER

Department of Mathematics Duke University Box 90320 Durham, NC 27708-0320 U.S.A. [email protected]

WENDELIN WERNER

Departement de Mathematiques Universite Paris-Sud Bat. 425 FR-91405 Orsay Cedex France wendelin. [email protected]

Received January 3, 2000

ODED SCHRAMM

Microsoft Research 1, Microsoft Way Redmond, WA 98052 U.S.A. [email protected]

values of brownian intersection exponents, i: half-plane ... · exponents from the sle6-exponents,...

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