random matrix theory - gbv · xviii detailed contents acknowledgements 227 references 227 11...
TRANSCRIPT
The Oxford Handbook of
Random Matrix Theory
Editors
Gemot Akemann, Jinho Balk and Philippe Di Francesco
w
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