Random Matrices, Random Processes and Integrable Systems

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.

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

From the reviews:

“The present volume consists of seven introductory articles originating from an intensive series of advanced courses given by the authors at the CRM in Montréal … . All articles are well-written by leading experts and the whole volume is an ideal starting point for anyone interested in this fascinating and modern topic at the intersection of mathematics and physics.” (G. Teschl, Monatshefte für Mathematik, Vol. 171 (3-4), September, 2013)

“This volume, written by the leading experts … provides a detailed look at many of the mathematical connections, ideas, directions, and methods that have come about since the early papers of Tracy and Widom. … this is a very nice collection of topics, especially for someone who wants to have all the RMT basics and also the basic computations and approaches that lead to other fields at hand. It should serve as a very valuable resource.” (Estelle L. Basor, Mathematical Reviews, February, 2013)

• Centre de Recherches Mathématiques, Université de Montréal, Montreal, Canada

  John Harnad

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