Opis: Combinatorics and Random Matrix Theory - Toufic Suidan, Percy Deift, Jinho Baik

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.* 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

Szczegóły: Combinatorics and Random Matrix Theory - Toufic Suidan, Percy Deift, Jinho Baik

Tytuł: Combinatorics and Random Matrix Theory
Autor: Toufic Suidan, Percy Deift, Jinho Baik
Wydawnictwo: American Mathematical Society
ISBN: 9780821848418
Rok wydania: 2016
Ilość stron: 461
Oprawa: Twarda

Recenzje: Combinatorics and Random Matrix Theory - Toufic Suidan, Percy Deift, Jinho Baik