The Oxford Handbook of
Random Matrix
Theory
Editors
Gemot Akemann, Jinho Balk and
Philippe
w
OXFORD
UNIVERSITY PRESS
Di Francesco
Detailed Contents
xxvii
List of Contributors
Introduction
Part I
1
Introduction and
G.
Akemann.J.
guide to the handbook
3
Abstract
2
3
Baik and P. Di Francesco
1.1
Random matrix
1.2
What is random matrix theory about?
5
1.3
Why is random matrix theory so successful?
Guide through this handbook
7
1.4
1.5
What is
1.6
Some
a
nutshell
covered in detail?
existing
3
8
11
introductory literature
12
Acknowledgements
13
References
14
History
O.
not
theory in
-
an
15
overview
Bohigas and
H. A. Weidenmuller
Abstract
15
15
2.1
Preface
2.2
Bohr's concept of the
2.3
Spectral properties
16
2.4
Data
21
2.5
Many-body theory
22
2.6
Chaos
23
2.7
Number
compound
nucleus
15
theory
Scattering theory
25
2.8
2.9
Replica trick and supersymmetry
29
25
2.10 Disordered solids
33
2.11
34
Interacting
fermions and field theory
Acknowledgements
35
References
35
Detailed Contents
xvi
Part II
3
Properties of random matrix theory
Symmetry Classes
43
M. R. Zirnbawr
Abstract
3.1
4
43
43
Introduction
threefold way
45
3.2
Dyson's
3.3
Symmetry classes of disordered fermions
52
3.4
Discussion
62
References
64
Spectral statistics of unitary ensembles
66
G. W. Anderson
5
Abstract
66
4.1
Introduction
66
4.2
The
orthogonal polynomial method: the setup
Examples: classical orthogonal polynomials
68
4.3
4.4
The
71
4.5
Cluster functions
fc-point correlation function
69
74
and Fredholm determinants
76
4.6
Gap probabilities
4.7
Resolvent kernels and
4.8
Spacings
83
References
84
Spectral statistics
Janossy densities
of orthogonal and
79
symplectic ensembles
86
M. Adler
Abstract
86
5.1
Introduction
86
5.2
Direct
5.3
Relations between
approach to the kernel
K# and
xjj',
polynomials
6
88
via
skew-orthogonal
96
References
101
Universality
A. B. J. Kuijlaars
103
Abstract
103
meaning of universality
Precise statement of universality
103
110
6.4
Unitary random matrix ensembles
Riemann-Hilbert method
6.5
Non-standard universality classes
126
Acknowledgements
130
References
131
6.1
6.2
6.3
Heuristic
105
115
Detailed, Contents
7
xvii
135
Supersymmetry
T. Guhr
8
Abstract
135
7.1
Generating functions
135
7.2
Supermathematics
137
7.3
Supersymmetric representation
142
7.4
Evaluation and structural
7.5
Circular ensembles and Colour-Flavour transformation
151
7.6
Concluding remarks
Acknowledgements
152
References
153
insights
148
153
Replica approach in random matrix theory
E.
Kanzieper
Abstract
155
8.1
Introduction
155
8.2
Early studies: heuristic approach
Integrable theory of replicas
Concluding remarks
Acknowledgements
8.3
8.4
to
replicas
159
165
173
174
References
9
155
174
Painleve transcendents
176
A. R. Its
Abstract
9.1
9.2
176
Introduction
176
Riemann-Hilbert representation
of the Painleve
functions
9.3
9.4
Asymptotic analysis
The Airy and the Sine kernels and
the Painleve functions
182
185
Acknowledgements
196
References
196
10 Random matrix
P.
178
of the Painleve functions
van
theory and integrable systems
198
Moerbeke
Abstract
10.1 Matrix
198
models, orthogonal polynomials,
and Kadomtsev-Petviashvili
198
10.2
(KP)
Multiple orthogonal polynomials
204
10.3
Critical diffusions
214
10.4 The Tacnode process
10.5
Kernels and ^-reduced KP
222
224
Detailed Contents
xviii
Acknowledgements
227
References
227
231
11 Determinantal point processes
A. Borodin
Abstract
231
11.1
Introduction
231
11.2
Generalities
232
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
244
11.9
Hermitian correlation kernels
245
Pfaffian point processes
246
Acknowledgements
247
References
247
11.3
11.10
12 Random matrix
spanning trees
representations of critical statistics
250
V. E. Kravtsov
250
Abstract
250
12.1
Introduction
12.2
Non-invariant Gaussian random matrix
multifractal
12.3
theory
with
252
eigenvectors
Invariant random matrix
theory (RMT)
with
log-square
confinement
12.4
12.5
12.6
12.7
12.8
12.9
13
254
and
self-unfolding
Self-unfolding
Unfolding and the spectral correlations
Ghost correlation dip in RMT and Hawking radiation
Invariant-noninvariant correspondence
Normalization anomaly, Luttinger liquid analogy
255
and the
263
in invariant RMT
not
Hawking temperature
258
259
261
Conclusions
267
Acknowledgements
268
References
268
270
Heavy-tailed random matrices
Z. Burda and J.
Jurkiewicz
Abstract
270
13.1
Introduction
270
13.2
Wigner-Levy matrices
13.3
Free random variables and free
272
Levy
matrices
278
Detailed Contents
xix
13.4
Heavy-tailed deformations
284
13.5
Summary
288
Acknowledgements
288
References
288
290
14 Phase transitions
G. M. Cicuta and L. G. Molinari
Abstract
290
14.1
Introduction
290
14.2
One-matrix models with
14.3
297
14.4
Eigenvalue matrix models
Complex matrix ensembles
14.5
Multi-matrix models
302
14.6
Matrix ensembles with preferred basis
303
References
306
polynomial potential
15 Two-matrix models and biorthogonal
292
300
polynomials
310
M. Bertola
Abstract
310
15.1
Introduction: chain-matrix models
310
15.2
The
Itzykson-Zuber Hermitian two-matrix model
Biorthogonal polynomials: Christoffel-Darboux identities
The spectral curve
Cauchy two-matrix models
311
References
327
15.3
15.4
15.5
16 Chain of matrices,
loop equations, and topological recursion
314
320
324
329
N. Orantin
329
Abstract
16.1
17
Introduction: what is
a
matrix
integral?
formal matrix integral
16.2
Convergent versus
16.3
Loop equations
16.4
Solution of the loop
329
330
334
equations
in the one-matrix model
337
16.5
Matrices
coupled in a chain plus external field
topological recursion
Acknowledgements
346
16.6
Generalization:
351
References
352
Unitary integrals and related matrix models
352
353
A. Morozov
353
Abstract
17.1
17.2
353
Introduction
Unitary integrals
and the Brezin-Gross-Witten model
355
Detailed
XX
17.3
Contents
Theory of the Harish-Chandxa-Itzykson-Zuber integral
Acknowledgements
361
References
373
376
18 Non-Hermitian ensembles
B. A. Khoruzhenko and
H.-J.
373
Sommers
Abstract
376
18.1
Introduction
376
18.2
Complex Ginibre ensemble
377
18.3
Random contractions
381
18.4
Complex elliptic ensemble
18.5
Real and
18.6
383
quaternion-real
Real and quaternion-real elliptic ensembles
Acknowledgements
386
References
396
Ginibre ensembles
393
396
398
19 Characteristic polynomials
£. Brezin and S. Hikami
Abstract
398
19.1
Introduction
398
19.2
Products of characteristic
19.3
Ratio of characteristic polynomials
polynomials
formula for an external
19.4
Duality
19.5
Fourier transform
source
U(s\, ...,Sk)
403
405
406
408
method
19.6
Replica
19.7
Intersection numbers of moduli space of curves
409
References
4-12
415
20 Beta ensembles
P.
J.
Forrester
Abstract.
415
Log-gas systems
Fokker-Planck equation and Calogero-Sutherland system
415
20.2
20.3
Matrix realization of j3 ensembles
425
20.4
Stochastic differential
429
20.1
21
399
equations
419
Acknowledgements
432
References
432
Wigner matrices
433
G. Ben Arous and A. Guionnet
Abstract
433
21.1
Introduction
433
21.2
Global
435
properties
Detailed Contents
xxi
21.3
Local properties in the bulk
441
21.4
Local
properties at the edge
Acknowledgements
446
References
450
probability theory
Speicher
452
450
22 Free
J?.
Abstract
452
22.1
Introduction
452
22.2
The moment method for several random matrices and the
22.3
concept of freeness
Basic definitions
22.4
Combinatorial
22.5
Free harmonic
22.6
Second-order freeness
45 2
456
theory of freeness
analysis
457
458
463
free
22.7
22.8
22.9
Operator-valued
probability theory
Further free-probabilistic aspects of random matrices
Operator algebraic aspects of free probability
Acknowledgements
463
References
469
23 Random banded and sparse matrices
T.
465
465
469
471
Spencer
Abstract
471
23.1
Introduction
23.2
Definition of random banded matrix
23.3
Density
23.4
Statistical mechanics and RBM
477
23.5
Eigenvectors of RBM
479
23.6
Random sparse matrices
Random Schrodinger on the Bethe lattice
484
Acknowledgments
486
References
486
23.7
24 Number
J.
P.
473
474
of random matrix
486
theory
theory
Keating
(RBM)
of states
Applications
Part III
471
ensembles
491
and N. C. Snaith
Abstract
491
24.1
Introduction
491
24.2
The number theoretical
24.3
Zero statistics
context
491
492
Detailed Contents
xxii
24.4
Values of the
Riemann zeta function
24.5
Values of I-functions
499
24.6
Further
502
areas
of interest
495
Acknowledgements
507
References
507
25 Random
permutations and related topics
510
G. Olshanski
Abstract
510
25.1
Introduction
25.2
The Ewens
510
measures,
virtual permutations, and the
Poisson-Dirichlet distributions
511
25.3
The Plancherel
518
25.4
The
measure
z-measures
and Schur measures
524
Acknowledgements
529
References
529
534
26 Enumeration of maps
J. Bouttier
26.1
26.2
Abstract
534
Introduction
534
Maps:
definitions
integrals to maps
degree distribution of planar maps
535
26.3
From matrix
538
26.4
The vertex
547
26.5
From matrix models to bijections
553
References
555
27 Knot theory and matrix integrals
P.
Zinn-Justin and J.-B.
557
Zuher
Abstract
557
27.1
Introduction and basic definitions
557
27.2
Matrix
27.3
Virtual knots
564
27.4
Coloured links
567
Acknowledgements
576
References
576
integrals, alternating links, and tangles
28 Multivariate statistics
559
578
N. El Karoui
Abstract
578
28.1
Introduction
578
28.2
Wishart distribution and normal
581
28.3
Extreme
584
theory
eigenvalues, Tracy-Widom laws
Detailed Contents
29
xxiii
28.4
Limiting spectral distribution results
590
28.5
Condusion
593
Acknowledgements
593
References
594
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
29.4
605
Higher expansion terms T\ and symplectic
invariants
615
Acknowledgements
617
References
617
30 Two-dimensional
quantum gravity
619
I. Rostov
Abstract
30.1
Introduction
30.2
Liouville
scaling
30.3
gravity
619
and
Knizhnik-Polyakov-Zamolodchikov
relation
620
Discretization of the
path integral
gravity and the
Ising model
over
metrics
Pure lattice
30.5
The
30.6
The
30.7
The six-vertex model
637
30.8
The q-state Potts model (0 < q < 4)
Solid-on-solid and ADE matrix models
637
References
638
0(n)
model
(-2
<
one-matrix model
625
30.4
30.9
31
619
n <
626
630
2)
String theory
632
638
641
M. Marino
Abstract
641
Introduction:
641
31.3
strings and matrices
A short survey of topological strings
The Drjkgraaf-Vafa correspondence
31.4
Matrix models and mirror symmetry
655
31.5
String theory, matrix quantum mechanics, and
31.1
31.2
644
650
related models
657
References
658
Detailed Contents
xxiv
32
661
Quantum chromodynamics
J. J. M. Verbaarschot
Abstract
661
Introduction
32.1
661
Quantum chromodynamics
32.2
and chiral random
matrix theory
663
Chiral random matrix
32.3
theory
at nonzero
chemical
potential
671
Applications to gauge degrees
Concluding remarks
Acknowledgments
32.4
32.5
of freedom
678
679
References
33
678
679
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
spectral
statistics
686
unitarity-preserving semiclassical
approximation
Analogy to the sigma model
Quantum graphs
33.4
A
33.5
33.6
References
34 Resonance
scattering of waves in chaotic systems
Introduction
34.2
Statistics
34.3
Correlation
34.4
Other characteristics and
at
matter
703
the fixed energy
705
properties
709
References
W.
703
703
34.1
C.
695
V. Savin
Abstract
35 Condensed
694
701
Fyodorov and D.
Y. V.
690
applications
716
720
physics
723
J. Beenakker
Abstract
723
35.1
Introduction
723
35.2
Quantum wires
724
35.3
dots
Quantum
35.4
Superconductors
737
References
741
729
Detailed Contents
36
xxv
Classical and quantum optics
C. W. J. Beenakker
744
Abstract
744
36.1
Introduction
744
36.2
Classical
745
36.3
Quantum optics
753
References
757
optics
eigenvalues of Wishart matrices: application
37 Extreme
to
entangled
bipartite system
S. N. Majumdar
759
Abstract
759
37.1
Introduction
37.2
Spectral properties of Wishart matrices: a brief summary
762
37.3
Entangled random pure state of a bipartite system
Minimum Eigenvalue distribution for quadratic matrices
766
Summary and conclusion
Acknowledgements
778
References
780
37.4
37.5
.
773
779
38 Random
P.
759
growth models
I. Ferrari and H. Spohn
782
Abstract
782
38.1
Growth models
38.2
How do random matrices
38.3
Multi-matrix models and line ensembles
786
38.4
Flat initial conditions
788
38.5
Growth models and last passage
791
38.6
Growth models and random
793
38.7
A
guide
to
782
appear?
percolation
tiling
the literature
795
References
39 Random matrices and
784
797
Laplacian growth
802
A. Zabrodin
Abstract
802
39.1
Introduction
39.2
Random matrices with
39.3
Exact relations at finite N
N
802
complex eigenvalues
limit
39.4
Large
39.5
The matrix model
804
808
811
as a
growth problem
818
Acknowledgments
822
References
822
xxvi
Detailed Contents
40 Financial applications of random matrix
J.-P.
41
theory:
a
short review
824
Bouchaud and M. Potters
Abstract
824
40.1
Introduction
824
40.2
Return statistics and
40.3
Random matrix theory: the bulk
833
40.4
Random matrix
839
40.5
Applications: cleaning correlation matrices
843
References
848
portfolio theory
theory:
the
827
edges
Asymptotic singular value distributions
in information theory
851
A. M. Tulino and S. Verdu
Abstract
851
851
41.2
The role of singular values in channel capacity
Transforms
41.3
Main results
856
References
868
41.1
42 Random matrix theory and ribonucleic acid
(RNA) folding
855
873
G. Vernizzi and H. Orland
43
Abstract
873
42.1
Introduction
873
42.2
A model for
877
42.3
Physical interpretation of the RN A matrix model
880
42.4
Large-N expansion
882
42.5
The
884
42.6
Numerical comparison
893
References
895
Complex networks
G. J. Rodgers and T. Nagao
898
RNA-folding
pseudoknotted homopolymer chain
Abstract
898
43.1
Introduction
898
43.2
Replica analysis
43.3
Local properties
References
Index
of scale free networks
900
909
911
912