The Oxford Handbook of

Random Matrix Theory

Editors

Gemot Akemann, Jinho Balk and Philippe Di Francesco

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OXFORDUNIVERSITY PRESS

Detailed Contents

List of Contributors xxvii

Part I Introduction

1 Introduction and guide to the handbook 3

G. Akemann.J. Baik and P. Di Francesco

Abstract 3

1.1 Random matrix theory in a nutshell 3

1.2 What is random matrix theory about? 5

1.3 Why is random matrix theory so successful? 7

1.4 Guide through this handbook 8

1.5 What is not covered in detail? 11

1.6 Some existing introductory literature 12

Acknowledgements 13

References 14

2 History - an overview 15

O. Bohigas and H. A. Weidenmuller

Abstract 15

2.1 Preface 15

2.2 Bohr's concept ofthe compound nucleus 15

2.3 Spectral properties 16

2.4 Data 21

2.5 Many-body theory 22

2.6 Chaos 23

2.7 Number theory 25

2.8 Scattering theory 25

2.9 Replica trick and supersymmetry 29

2.10 Disordered solids 33

2.11 Interacting fermions and field theory 34

Acknowledgements 35

References 35

xvi Detailed Contents

Part II Properties ofrandom matrix theory

3 Symmetry Classes 43

M. R. Zirnbawr

Abstract 43

3.1 Introduction 43

3.2 Dyson's threefold way 45

3.3 Symmetry classes of disordered fermions 52

3.4 Discussion 62

References 64

4 Spectral statistics of unitary ensembles 66

G. W. Anderson

Abstract 66

4.1 Introduction 66

4.2 The orthogonal polynomial method: the setup 68

4.3 Examples: classical orthogonal polynomials 69

4.4 The fc-point correlation function 71

4.5 Cluster functions 74

4.6 Gap probabilities and Fredholm determinants 76

4.7 Resolvent kernels and Janossy densities 79

4.8 Spacings 83

References 84

5 Spectral statistics oforthogonal and symplectic ensembles 86

M. Adler

Abstract 86

5.1 Introduction 86

5.2 Direct approach to the kernel 88

5.3 Relations between K# and xjj', via skew-orthogonalpolynomials 96

References 101

6 Universality 103

A. B. J. KuijlaarsAbstract 103

6.1 Heuristic meaning ofuniversality 103

6.2 Precise statement of universality 105

6.3 Unitary random matrix ensembles 110

6.4 Riemann-Hilbert method 115

6.5 Non-standard universality classes 126

Acknowledgements 130

References 131

Detailed, Contents xvii

7 Supersymmetry 135

T. Guhr

Abstract 135

7.1 Generating functions 135

7.2 Supermathematics 137

7.3 Supersymmetric representation 142

7.4 Evaluation and structural insights 148

7.5 Circular ensembles and Colour-Flavour transformation 151

7.6 Concluding remarks 152

Acknowledgements 153

References 153

8 Replica approach in random matrix theory 155

E. KanzieperAbstract 155

8.1 Introduction 155

8.2 Early studies: heuristic approach to replicas 159

8.3 Integrable theory of replicas 165

8.4 Concluding remarks 173

Acknowledgements 174

References 174

9 Painleve transcendents 176

A. R. Its

Abstract 176

9.1 Introduction 176

9.2 Riemann-Hilbert representation of the Painleve

functions 178

9.3 Asymptotic analysis ofthe Painleve functions 182

9.4 The Airy and the Sine kernels and the Painleve functions 185

Acknowledgements 196

References 196

10 Random matrix theory and integrable systems 198

P. van Moerbeke

Abstract 198

10.1 Matrix models, orthogonal polynomials,and Kadomtsev-Petviashvili (KP) 198

10.2 Multiple orthogonal polynomials 204

10.3 Critical diffusions 214

10.4 The Tacnode process 222

10.5 Kernels and ^-reduced KP 224

xviii Detailed Contents

Acknowledgements 227

References 227

11 Determinantal point processes 231

A. Borodin

Abstract 231

11.1 Introduction 231

11.2 Generalities 232

11.3 Loop-free Markov chains 234

11.4 Measures given by products of determinants 235

11.5 L-ensembles 240

11.6 Fock space 241

11.7 Dimer models 244

11.8 Uniform spanning trees 244

11.9 Hermitian correlation kernels 245

11.10 Pfaffian point processes 246

Acknowledgements 247

References 247

12 Random matrix representations ofcritical statistics 250

V. E. Kravtsov

Abstract 250

12.1 Introduction 250

12.2 Non-invariant Gaussian random matrix theory with

multifractal eigenvectors 252

12.3 Invariant random matrix theory (RMT) with log-squareconfinement 254

12.4 Self-unfolding and not self-unfolding in invariant RMT 255

12.5 Unfolding and the spectral correlations 258

12.6 Ghost correlation dip in RMT and Hawking radiation 259

12.7 Invariant-noninvariant correspondence 261

12.8 Normalization anomaly, Luttinger liquid analogyand the Hawking temperature 263

12.9 Conclusions 267

Acknowledgements 268

References 268

13 Heavy-tailed random matrices 270

Z. Burda and J. JurkiewiczAbstract 270

13.1 Introduction 270

13.2 Wigner-Levy matrices 272

13.3 Free random variables and free Levy matrices 278

Detailed Contents xix

13.4 Heavy-tailed deformations 284

13.5 Summary 288

Acknowledgements 288

References 288

14 Phase transitions 290

G. M. Cicuta and L. G. Molinari

Abstract 290

14.1 Introduction 290

14.2 One-matrix models with polynomial potential 292

14.3 Eigenvalue matrix models 297

14.4 Complex matrix ensembles 300

14.5 Multi-matrix models 302

14.6 Matrix ensembles with preferred basis 303

References 306

15 Two-matrix models and biorthogonal polynomials 310

M. Bertola

Abstract 310

15.1 Introduction: chain-matrix models 310

15.2 The Itzykson-Zuber Hermitian two-matrix model 311

15.3 Biorthogonal polynomials: Christoffel-Darboux identities 314

15.4 The spectral curve 320

15.5 Cauchy two-matrix models 324

References 327

16 Chain ofmatrices, loop equations, and topological recursion 329

N. Orantin

Abstract 329

16.1 Introduction: what is a matrix integral? 329

16.2 Convergent versus formal matrix integral 330

16.3 Loop equations 334

16.4 Solution of the loop equations in the one-matrix model 337

16.5 Matrices coupled in a chain plus external field 346

16.6 Generalization: topological recursion 351

Acknowledgements 352

References 352

17 Unitary integrals and related matrix models 353

A. Morozov

Abstract 353

17.1 Introduction 353

17.2 Unitary integrals and the Brezin-Gross-Witten model 355

XX Detailed Contents

17.3 Theory of the Harish-Chandxa-Itzykson-Zuberintegral 361

Acknowledgements 373

References 373

18 Non-Hermitian ensembles 376

B. A. Khoruzhenko and H.-J. Sommers

Abstract 376

18.1 Introduction 376

18.2 Complex Ginibre ensemble 377

18.3 Random contractions 381

18.4 Complex elliptic ensemble 383

18.5 Real and quaternion-real Ginibre ensembles 386

18.6 Real and quaternion-real elliptic ensembles 393

Acknowledgements 396

References 396

19 Characteristic polynomials 398

£. Brezin and S. Hikami

Abstract 398

19.1 Introduction 398

19.2 Products of characteristic polynomials 399

19.3 Ratio ofcharacteristic polynomials 403

19.4 Duality formula for an external source 405

19.5 Fourier transform U(s\, ...,Sk) 406

19.6 Replica method 408

19.7 Intersection numbers ofmoduli space of curves 409

References 4-12

20 Beta ensembles 415

P. J. Forrester

Abstract. 415

20.1 Log-gas systems 415

20.2 Fokker-Planck equation and Calogero-Sutherland system 419

20.3 Matrix realization of j3 ensembles 425

20.4 Stochastic differential equations 429

Acknowledgements 432

References 432

21 Wigner matrices 433

G. Ben Arous and A. Guionnet

Abstract 433

21.1 Introduction 433

21.2 Global properties 435

Detailed Contents xxi

21.3 Local properties in the bulk 441

21.4 Local properties at the edge 446

Acknowledgements 450

References 450

22 Free probability theory 452

J?. SpeicherAbstract 452

22.1 Introduction 452

22.2 The moment method for several random matrices and the

concept offreeness 452

22.3 Basic definitions 456

22.4 Combinatorial theory of freeness 457

22.5 Free harmonic analysis 458

22.6 Second-order freeness 463

22.7 Operator-valued free probability theory 463

22.8 Further free-probabilistic aspects ofrandom matrices 465

22.9 Operator algebraic aspects offree probability 465

Acknowledgements 469

References 469

23 Random banded and sparse matrices 471

T. SpencerAbstract 471

23.1 Introduction 471

23.2 Definition of random banded matrix (RBM) ensembles 473

23.3 Density of states 474

23.4 Statistical mechanics and RBM 477

23.5 Eigenvectors of RBM 479

23.6 Random sparse matrices 484

23.7 Random Schrodinger on the Bethe lattice 486

Acknowledgments 486

References 486

Part III Applications ofrandom matrix theory

24 Number theory 491

J. P. Keating and N. C. Snaith

Abstract 491

24.1 Introduction 491

24.2 The number theoretical context 491

24.3 Zero statistics 492

xxii Detailed Contents

24.4 Values of the Riemann zeta function 495

24.5 Values ofI-functions 499

24.6 Further areas ofinterest 502

Acknowledgements 507

References 507

25 Random permutations and related topics 510

G. Olshanski

Abstract 510

25.1 Introduction 510

25.2 The Ewens measures, virtual permutations, and the

Poisson-Dirichlet distributions 511

25.3 The Plancherel measure 518

25.4 The z-measures and Schur measures 524

Acknowledgements 529

References 529

26 Enumeration of maps 534

J. Bouttier

Abstract 534

26.1 Introduction 534

26.2 Maps: definitions 535

26.3 From matrix integrals to maps 538

26.4 The vertex degree distribution ofplanar maps 547

26.5 From matrix models to bijections 553

References 555

27 Knot theory and matrix integrals 557

P. Zinn-Justin andJ.-B. Zuher

Abstract 557

27.1 Introduction and basic definitions 557

27.2 Matrix integrals, alternating links, and tangles 559

27.3 Virtual knots 564

27.4 Coloured links 567

Acknowledgements 576

References 576

28 Multivariate statistics 578

N. El Karoui

Abstract 578

28.1 Introduction 578

28.2 Wishart distribution and normal theory 581

28.3 Extreme eigenvalues, Tracy-Widom laws 584

Detailed Contents xxiii

28.4 Limiting spectral distribution results 590

28.5 Condusion 593

Acknowledgements 593

References 594

29 Algebraic geometry and matrix models 597

I. O. Chekhov

Abstract 597

29.1 Introduction 597

29.2 Moduli spaces and matrix models 598

29.3 The planar term ^ and Witten-Dijkgraaf-Verlinde-Verlinde 605

29.4 Higher expansion terms T\ and symplecticinvariants 615

Acknowledgements 617

References 617

30 Two-dimensional quantum gravity 619

I. Rostov

Abstract 619

30.1 Introduction 619

30.2 Liouville gravity and Knizhnik-Polyakov-Zamolodchikovscaling relation 620

30.3 Discretization ofthe path integral over metrics 625

30.4 Pure lattice gravity and the one-matrix model 626

30.5 The Ising model 630

30.6 The 0(n) model (-2 < n < 2) 632

30.7 The six-vertex model 637

30.8 The q-state Potts model (0 < q < 4) 637

30.9 Solid-on-solid and ADE matrix models 638

References 638

31 String theory 641

M. Marino

Abstract 641

31.1 Introduction: strings and matrices 641

31.2 A short survey of topological strings 644

31.3 The Drjkgraaf-Vafa correspondence 650

31.4 Matrix models and mirror symmetry 655

31.5 String theory, matrix quantum mechanics, and

related models 657

References 658

xxiv Detailed Contents

32 Quantum chromodynamics 661

J. J. M. Verbaarschot

Abstract 661

32.1 Introduction 661

32.2 Quantum chromodynamics and chiral random

matrix theory 663

32.3 Chiral random matrix theory at nonzero chemical

potential 671

32.4 Applications to gauge degrees offreedom 678

32.5 Concluding remarks 678

Acknowledgments 679

References 679

33 Quantum chaos and quantum graphs 683

S. Mutter and M. Sieher

Abstract 683

33.1 Introduction 683

33.2 Classical chaos 684

33.3 Gutzwiller's trace formula and spectralstatistics 686

33.4 A unitarity-preserving semiclassical

approximation 690

33.5 Analogy to the sigma model 694

33.6 Quantum graphs 695

References 701

34 Resonance scattering ofwaves in chaotic systems 703

Y. V. Fyodorov and D. V. Savin

Abstract 703

34.1 Introduction 703

34.2 Statistics at the fixed energy 705

34.3 Correlation properties 709

34.4 Other characteristics and applications 716

References 720

35 Condensed matter physics 723

C. W. J. Beenakker

Abstract 723

35.1 Introduction 723

35.2 Quantum wires 724

35.3 Quantum dots 729

35.4 Superconductors 737

References 741

Detailed Contents xxv

36 Classical and quantum optics 744

C. W. J. Beenakker

Abstract 744

36.1 Introduction 744

36.2 Classical optics 745

36.3 Quantum optics 753

References 757

37 Extreme eigenvalues of Wishart matrices: application to entangledbipartite system 759

S. N. MajumdarAbstract 759

37.1 Introduction.

759

37.2 Spectral properties of Wishart matrices: a brief summary 762

37.3 Entangled random pure state of a bipartite system 766

37.4 Minimum Eigenvalue distribution for quadratic matrices 773

37.5 Summary and conclusion 778

Acknowledgements 779

References 780

38 Random growth models 782

P. I. Ferrari and H. SpohnAbstract 782

38.1 Growth models 782

38.2 How do random matrices appear? 784

38.3 Multi-matrix models and line ensembles 786

38.4 Flat initial conditions 788

38.5 Growth models and last passage percolation 791

38.6 Growth models and random tiling 793

38.7 A guide to the literature 795

References 797

39 Random matrices and Laplacian growth 802

A. Zabrodin

Abstract 802

39.1 Introduction 802

39.2 Random matrices with complex eigenvalues 804

39.3 Exact relations at finite N 808

39.4 Large N limit 811

39.5 The matrix model as a growth problem 818

Acknowledgments 822

References 822

xxvi Detailed Contents

40 Financial applications of random matrix theory: a short review 824

J.-P. Bouchaud and M. Potters

Abstract 824

40.1 Introduction 824

40.2 Return statistics and portfolio theory 827

40.3 Random matrix theory: the bulk 833

40.4 Random matrix theory: the edges 839

40.5 Applications: cleaning correlation matrices 843

References 848

41 Asymptotic singular value distributions in information theory 851

A. M. Tulino and S. Verdu

Abstract 851

41.1 The role of singular values in channel capacity 851

41.2 Transforms 855

41.3 Main results 856

References 868

42 Random matrix theory and ribonucleic acid (RNA) folding 873

G. Vernizzi and H. Orland

Abstract 873

42.1 Introduction 873

42.2 A model for RNA-folding 877

42.3 Physical interpretation of the RNA matrix model 880

42.4 Large-N expansion 882

42.5 The pseudoknotted homopolymer chain 884

42.6 Numerical comparison 893

References 895

43 Complex networks 898

G. J. Rodgers and T. NagaoAbstract 898

43.1 Introduction 898

43.2 Replica analysis of scale free networks 900

43.3 Local properties 909

References 911

Index 912