Random Matrix Theory, Interacting Particle Systems, and Integrable Systems

Mathematical Sciences Research Institute

doi:10.1017/9781139942287

Random matrix theory is at the intersection of linear algebra, probability theory and integrable systems, and has a wide range of applications in physics, engineering, multivariate statistics and beyond. This volume is based on a Fall 2010 MSRI program which generated the solution of long-standing questions on universalities of Wigner matrices and beta-ensembles and opened new research directions especially in relation to the KPZ universality class of interacting particle systems and low-rank perturbations. The book contains review articles and research contributions on all these topics, in addition to other core aspects of random matrix theory such as integrability and free probability theory. It will give both established and new researchers insights into the most recent advances in the field and the connections among many subfields.

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Contents

Frontmatter
pp i-vi
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Contents
pp vii-viii
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Preface
pp ix-x
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Universality conjecture for all Airy, sine and Bessel kernels in the complex plane
pp 1-24
- By Gernot Akemann, Michael J. Phillips
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On a relationship between high rank cases and rank one cases of Hermitian random matrix models with external
pp 25-38
- By Jinho Baik, Dong Wang
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Riemann–Hilbert approach to the six-vertex model
pp 39-56
- By Pavel Bleher, Karl Liechty
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CLT for spectra of submatrices of Wigner random matrices, II: Stochastic evolution
pp 57-70
- By Alexei Borodin
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Critical asymptotic behavior for the Korteweg–de Vries equation and in random matrix theory
pp 71-92
- By Tom Claeys, Tamara Grava
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On the asymptotics of a Toeplitz determinant with singularities
pp 93-146
- By Percy Deift, Alexander Its, Igor Krasovsky
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Asymptotic analysis of the two-matrix model with a quartic potential
pp 147-162
- By Maurice Duits, Arno B. J. Kuijlaars, Man Yue Mo
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Conservation laws of random matrix theory
pp 163-198
- By Nicholas M. Ercolani
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Asymptotics of spacing distributions 50 years later
pp 199-222
- By Peter J. Forrester
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Applications of random matrix theory for sensor array imaging with measurement noise
pp 223-246
- By Josselin Garnier, Knut Sølna
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Convolution symmetries of integrable hierarchies, matrix models and τ-functions
pp 247-276
- By J. Harnad, A. Yu. Orlov
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Universality limits via “old style” analysis
pp 277-292
- By Doron S. Lubinsky
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Fluctuations and large deviations of some perturbed random matrices
pp 293-308
- By Mylène Maïda
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Three lectures on free probability
pp 309-384
- By Jonathan Novak
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Whittaker functions and relatedstochastic processes
pp 385-410
- By Neil O’ connell
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How long does it take to compute the eigenvalues of a random symmetric matrix?
pp 411-442
- By Christian W. Pfrang, Percy Deift, Govind Menon
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Exact solutions of the Kardar–Parisi–Zhang equation and weak universality for directed random polymers
pp 443-450
- By Jeremy Quastel
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Replica analysis of the one-dimensional KPZ equation
pp 451-462
- By Tomohiro Sasamoto
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Asymptotic expansions for β matrix models and their applications to the universality conjecture
pp 463-482
- By Mariya Shcherbina
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KPZ scaling theory and the semidiscrete directed polymer model
pp 483-494
- By Herbert Spohn
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Experimental Realization Of Tracy–Widom Distributions And Beyond: Kpz Interfaces In Turbulent Liquid Crystal
pp 495-508
- By Kazumasa A. Takeuchi
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Random matrices: the four-moment theorem for Wigner ensembles
pp 509-528
- By Terence Tao, Van Vu
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Random Matrix Theory, Interacting Particle Systems, and Integrable Systems

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Contents

Frontmatter
pp i-vi

Contents
pp vii-viii

Preface
pp ix-x

Universality conjecture for all Airy, sine and Bessel kernels in the complex plane
pp 1-24

On a relationship between high rank cases and rank one cases of Hermitian random matrix models with external
pp 25-38

Riemann–Hilbert approach to the six-vertex model
pp 39-56

CLT for spectra of submatrices of Wigner random matrices, II: Stochastic evolution
pp 57-70

Critical asymptotic behavior for the Korteweg–de Vries equation and in random matrix theory
pp 71-92

On the asymptotics of a Toeplitz determinant with singularities
pp 93-146

Asymptotic analysis of the two-matrix model with a quartic potential
pp 147-162

Conservation laws of random matrix theory
pp 163-198

Asymptotics of spacing distributions 50 years later
pp 199-222

Applications of random matrix theory for sensor array imaging with measurement noise
pp 223-246

Convolution symmetries of integrable hierarchies, matrix models and τ-functions
pp 247-276

Universality limits via “old style” analysis
pp 277-292

Fluctuations and large deviations of some perturbed random matrices
pp 293-308

Three lectures on free probability
pp 309-384

Whittaker functions and relatedstochastic processes
pp 385-410

How long does it take to compute the eigenvalues of a random symmetric matrix?
pp 411-442

Exact solutions of the Kardar–Parisi–Zhang equation and weak universality for directed random polymers
pp 443-450

Replica analysis of the one-dimensional KPZ equation
pp 451-462

Asymptotic expansions for β matrix models and their applications to the universality conjecture
pp 463-482

KPZ scaling theory and the semidiscrete directed polymer model
pp 483-494

Experimental Realization Of Tracy–Widom Distributions And Beyond: Kpz Interfaces In Turbulent Liquid Crystal
pp 495-508

Random matrices: the four-moment theorem for Wigner ensembles
pp 509-528

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Random Matrix Theory, Interacting Particle Systems, and Integrable Systems

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