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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.
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.
Tau functions are a central tool in the modern theory of integrable
systems. This volume provides a thorough introduction, starting
from the basics and extending to recent research results. It covers
a wide range of applications, including generating functions for
solutions of integrable hierarchies, correlation functions in the
spectral theory of random matrices and combinatorial generating
functions for enumerative geometrical and topological invariants. A
self-contained summary of more advanced topics needed to understand
the material is provided, as are solutions and hints for the
various exercises and problems that are included throughout the
text to enrich the subject matter and engage the reader. Building
on knowledge of standard topics in undergraduate mathematics and
basic concepts and methods of classical and quantum mechanics, this
monograph is ideal for graduate students and researchers who wish
to become acquainted with the full range of applications of the
theory of tau functions.
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