Combinatorics and random matrix theory
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Bibliographic Information
Combinatorics and random matrix theory
（Graduate studies in mathematics, v. 172）
American Mathematical Society, c2016
Available at / 36 libraries
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Library, Research Institute for Mathematical Sciences, Kyoto University数研
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Note
Includes bibliographical references (p. 445-458) and index
Description and Table of Contents
Description
Over the last fifteen years a variety of problems in combinatorics has been solved in terms of random matrix theory. More precisely, the situation is as follows: the problems at hand are probabilistic in nature and, in an appropriate scaling limit, it turns out that certain key quantities associated with these problems behave statistically like the eigenvalues of a (large) random matrix. Said differently, random matrix theory provides a ``stochastic special function theory'' for a broad and growing class of problems in combinatorics. The goal of this book is to analyze in detail two key examples of this phenomenon, viz., Ulam's problem for increasing subsequences of random permutations and domino tilings of the Aztec diamond. Other examples are also described along the way, but in less detail.
Techniques from many different areas in mathematics are needed to analyze these problems. These areas include combinatorics, probability theory, functional analysis, complex analysis, and the theory of integrable systems. The book is self-contained, and along the way we develop enough of the theory we need from each area that a general reader with, say, two or three years experience in graduate school can learn the subject directly from the text.
Table of Contents
Introduction
Poissonization and de-Poissonization
Permutations and Young tableaux
Bounds of the expected value of $\ell_N$
Orthogonal polynomials, Riemann-Hilbert problems, and Toeplitz matrices
Random matrix theory
Toeplitz determinant formula
Fredholm determinant formula
Asymptotic results
Schur measure and directed last passage percolation
Determinantal point processes
Tiling of the Aztec diamond
The Dyson process and Brownian Dyson process
Theory of trace class operators and Fredholm determinants
Steepest-descent method for the asymptotic evaluation of integrals in the complex plane
Basic results of stochastic calculus
Bibliography
Index
by "Nielsen BookData"
