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Physics - Theoretical, Mathematical & Computational Physics | Random Matrices, Random Processes and Integrable Systems

Random Matrices, Random Processes and Integrable Systems

Harnad, John (Ed.)

2011, XVIII, 526 p.

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  • Provides an in-depth examination of random matrices with applications over a vast variety of domains, including multivariate statistics, random growth models, and many others
  • Applies the theory of integrable systems, a source of powerful analytic methods, to the solution of fundamental problems in random systems and processes
  • Features an interdisciplinary approach that sheds new light on a dynamic topic of current research
  • Explains and develops the phenomenon of "universality," in particular, the occurrence of the Tracy-Widom distribution for eigenvalues at the "edge of the spectrum," in the longest increasing subsequence of a random permutation and a variety of critical phenomena in the double scaling limit

This book explores the remarkable connections between two domains that, a priori, seem unrelated: Random matrices (together with associated random processes) and integrable systems. The relations between random matrix models and the theory of classical integrable systems have long been studied. These appear mainly in the deformation theory, when parameters characterizing the measures or the domain of localization of the eigenvalues are varied. The resulting differential equations determining the partition function and correlation functions are, remarkably, of the same type as certain equations appearing in the theory of integrable systems. They may be analyzed effectively through methods based upon the Riemann-Hilbert problem of analytic function theory and by related approaches to the study of nonlinear asymptotics in the large N limit. Associated with studies of matrix models are certain stochastic processes, the "Dyson processes", and their continuum diffusion limits, which govern the spectrum in random matrix ensembles, and may also be studied by related methods.

Random Matrices, Random Processes and Integrable Systems provides an in-depth examination of random matrices with applications over a vast variety of domains, including multivariate statistics, random growth models, and many others. Leaders in the field apply the theory of integrable systems to the solution of fundamental problems in random systems and processes using an interdisciplinary approach that sheds new light on a dynamic topic of current research.

Content Level » Research

Keywords » Riemann-Hibert method - integrable systems - nonlinear steepest descent - random growth models - random matrices - random processes - random sequences

Related subjects » Probability Theory and Stochastic Processes - Theoretical, Mathematical & Computational Physics

Table of contents 

Introduction by John Harnad

Part I Random Matrices, Random Processes and Integrable Models
Chapter 1 Random and Integrable Models in Mathematics and Physics by Pierre van Moerbeke
Chapter 2 Integrable Systems, Random Matrices, and Random Processes by Mark Adler 

Part II Random Matrices and Applications
Chapter 3 Integral Operators in Random Matrix Theory by Harold Widom
Chapter 4 Lectures on Random Matrix Models by Pavel M. Bleher
Chapter 5 Large N Asymptotics in Random Matrices by Alexander R. Its
Chapter 6 Formal Matrix Integrals and Combinatorics of Maps by B. Eynard
Chapter 7 Application of Random Matrix Theory to Multivariate Statistics by Momar Dieng and Craig A. Tracy

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