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.

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