Random matrices, random processes and integrable systems

✍ Scribed by Pierre van Moerbeke (auth.), John Harnad (eds.)

• Publisher: Springer-Verlag New York
• Year: 2011
• Tongue: English
• Leaves: 544
• Series: CRM Series in Mathematical Physics
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter....Pages i-xviii
Front Matter....Pages 1-1
Random and Integrable Models in Mathematics and Physics....Pages 3-130
Integrable Systems, Random Matrices, and Random Processes....Pages 131-225
Front Matter....Pages 227-227
Integral Operators in Random Matrix Theory....Pages 229-249
Lectures on Random Matrix Models....Pages 251-349
Large N Asymptotics in Random Matrices....Pages 351-413
Formal Matrix Integrals and Combinatorics of Maps....Pages 415-442
Application of Random Matrix Theory to Multivariate Statistics....Pages 443-507
Back Matter....Pages 509-524

✦ Subjects

Theoretical, Mathematical and Computational Physics;Probability Theory and Stochastic Processes