Random Matrices, Random Processes and Integrable Systems

  • John Harnad

Part of the CRM Series in Mathematical Physics book series (CRM)

Table of contents

  1. Front Matter
    Pages i-xviii
  2. Random Matrices, Random Processes and Integrable Models

    1. Front Matter
      Pages 1-1
  3. Random Matrices and Applications

    1. Front Matter
      Pages 227-227
    2. Harold Widom
      Pages 229-249
    3. Pavel M. Bleher
      Pages 251-349
    4. Alexander R. Its
      Pages 351-413
    5. Momar Dieng, Craig A. Tracy
      Pages 443-507
  4. Back Matter
    Pages 509-524

About this book

Introduction

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.

Keywords

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

Editors and affiliations

  • John Harnad
    • 1
  1. 1.Centre de Recherches MathématiquesUniversité de MontréalMontrealCanada

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4419-9514-8
  • Copyright Information Springer Science+Business Media, LLC 2011
  • Publisher Name Springer, New York, NY
  • eBook Packages Physics and Astronomy
  • Print ISBN 978-1-4419-9513-1
  • Online ISBN 978-1-4419-9514-8
  • About this book