Combinatorics and random matrix theory

著者

• Baik, J.
• Deift, Percy
• Suidan, Toufic Mubadda

書誌事項

Combinatorics and random matrix theory

Jinho Baik, Percy Deift, Toufic Suidan

（Graduate studies in mathematics, v. 172）

American Mathematical Society, c2016

注記

Includes bibliographical references (p. 445-458) and index

内容説明・目次

内容説明

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.

目次

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

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詳細情報

• NII書誌ID(NCID): BB21533770
• ISBN:
  • 9780821848418
• LCCN: 2015051274
• 出版国コード: us
• タイトル言語コード: eng
• 本文言語コード: eng
• 出版地: Providence, R.I.
• ページ数/冊数: xi, 461 p.
• 大きさ: 26 cm
• 分類:
  • LCC : QA188
  • DC23 : 511/.6
• 件名:
  • LCSH : Random matrices
  • LCSH : Combinatorial analysis
  • FREE : Combinatorics -- Enumerative combinatorics -- Exact enumeration problems, generating functions. msc
  • FREE : Linear and multilinear algebra; matrix theory -- Special matrices -- Random matrices. msc
  • FREE : Special functions (33-XX deals with the properties of functions as functions) -- Other special functions -- Painlevé-type functions. msc
  • FREE : Partial differential equations -- Equations of mathematical physics and other areas of application -- Riemann-Hilbert problems. msc
  • FREE : Approximations and expansions -- Approximations and expansions -- Asymptotic approximations, asymptotic expansions (steepest descent, etc.). msc
  • FREE : Operator theory -- Special classes of linear operators -- Toeplitz operators, Hankel operators, Wiener-Hopf operators. msc
  • FREE : Convex and discrete geometry -- Discrete geometry -- Tilings in $2$ dimensions. msc
  • FREE : Probability theory and stochastic processes -- Probability theory on algebraic and topological structures -- Random matrices (probabilistic aspects; for algebraic aspects see 15B52). msc
  • FREE : Probability theory and stochastic processes -- Special processes -- Interacting random processes; statistical mechanics type models; percolation theory. msc
  • FREE : Statistical mechanics, structure of matter -- Time-dependent statistical mechanics (dynamic and nonequilibrium) -- Exactly solvable dynamic models. msc
• 親書誌ID:
  • BA20542214