Combinatorics And Random Matrix Theory

Author : Jinho Baik,Percy Deift,Toufic Suidan
Publisher : American Mathematical Soc.
Page : 461 pages
File Size : 43,7 Mb
Release : 2016-06-22
Category : Combinatorial analysis
ISBN : 9780821848418

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Combinatorics and Random Matrix Theory by Jinho Baik,Percy Deift,Toufic Suidan Pdf

Over the last fifteen years a variety of problems in combinatorics have 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.

Author : Alexei Borodin,Ivan Corwin,Alice Guionnet
Publisher : American Mathematical Soc.
Page : 498 pages
File Size : 44,7 Mb
Release : 2019-10-30
Category : Education
ISBN : 9781470452803

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Random Matrices by Alexei Borodin,Ivan Corwin,Alice Guionnet Pdf

Random matrix theory has many roots and many branches in mathematics, statistics, physics, computer science, data science, numerical analysis, biology, ecology, engineering, and operations research. This book provides a snippet of this vast domain of study, with a particular focus on the notations of universality and integrability. Universality shows that many systems behave the same way in their large scale limit, while integrability provides a route to describe the nature of those universal limits. Many of the ten contributed chapters address these themes, while others touch on applications of tools and results from random matrix theory. This book is appropriate for graduate students and researchers interested in learning techniques and results in random matrix theory from different perspectives and viewpoints. It also captures a moment in the evolution of the theory, when the previous decade brought major break-throughs, prompting exciting new directions of research.

Author : Pavel Bleher,Alexander Its
Publisher : Cambridge University Press
Page : 454 pages
File Size : 52,7 Mb
Release : 2001-06-04
Category : Mathematics
ISBN : 0521802091

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Random Matrix Models and Their Applications by Pavel Bleher,Alexander Its Pdf

Expository articles on random matrix theory emphasizing the exchange of ideas between the physical and mathematical communities.

Author : Alice Guionnet
Publisher : Springer
Page : 294 pages
File Size : 55,7 Mb
Release : 2009-04-20
Category : Mathematics
ISBN : 9783540698975

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Large Random Matrices: Lectures on Macroscopic Asymptotics by Alice Guionnet Pdf

Random matrix theory has developed in the last few years, in connection with various fields of mathematics and physics. These notes emphasize the relation with the problem of enumerating complicated graphs, and the related large deviations questions. Such questions are also closely related with the asymptotic distribution of matrices, which is naturally defined in the context of free probability and operator algebra. The material of this volume is based on a series of nine lectures given at the Saint-Flour Probability Summer School 2006. Lectures were also given by Maury Bramson and Steffen Lauritzen.

Author : Terence Tao
Publisher : American Mathematical Society
Page : 296 pages
File Size : 49,6 Mb
Release : 2023-08-24
Category : Mathematics
ISBN : 9781470474591

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Topics in Random Matrix Theory by Terence Tao Pdf

The field of random matrix theory has seen an explosion of activity in recent years, with connections to many areas of mathematics and physics. However, this makes the current state of the field almost too large to survey in a single book. In this graduate text, we focus on one specific sector of the field, namely the spectral distribution of random Wigner matrix ensembles (such as the Gaussian Unitary Ensemble), as well as iid matrix ensembles. The text is largely self-contained and starts with a review of relevant aspects of probability theory and linear algebra. With over 200 exercises, the book is suitable as an introductory text for beginning graduate students seeking to enter the field.

Author : Greg W. Anderson,Alice Guionnet,Ofer Zeitouni
Publisher : Cambridge University Press
Page : 507 pages
File Size : 45,9 Mb
Release : 2010
Category : Mathematics
ISBN : 9780521194525

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An Introduction to Random Matrices by Greg W. Anderson,Alice Guionnet,Ofer Zeitouni Pdf

A rigorous introduction to the basic theory of random matrices designed for graduate students with a background in probability theory.

Author : Giacomo Livan,Marcel Novaes,Pierpaolo Vivo
Publisher : Springer
Page : 124 pages
File Size : 48,6 Mb
Release : 2018-01-16
Category : Science
ISBN : 9783319708850

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Introduction to Random Matrices by Giacomo Livan,Marcel Novaes,Pierpaolo Vivo Pdf

Modern developments of Random Matrix Theory as well as pedagogical approaches to the standard core of the discipline are surprisingly hard to find in a well-organized, readable and user-friendly fashion. This slim and agile book, written in a pedagogical and hands-on style, without sacrificing formal rigor fills this gap. It brings Ph.D. students in Physics, as well as more senior practitioners, through the standard tools and results on random matrices, with an eye on most recent developments that are not usually covered in introductory texts. The focus is mainly on random matrices with real spectrum.The main guiding threads throughout the book are the Gaussian Ensembles. In particular, Wigner’s semicircle law is derived multiple times to illustrate several techniques (e.g., Coulomb gas approach, replica theory).Most chapters are accompanied by Matlab codes (stored in an online repository) to guide readers through the numerical check of most analytical results.

Author : Van H. Vu
Publisher : American Mathematical Society
Page : 176 pages
File Size : 42,6 Mb
Release : 2014-07-16
Category : Mathematics
ISBN : 9780821894712

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Modern Aspects of Random Matrix Theory by Van H. Vu Pdf

The theory of random matrices is an amazingly rich topic in mathematics. Random matrices play a fundamental role in various areas such as statistics, mathematical physics, combinatorics, theoretical computer science, number theory and numerical analysis. This volume is based on lectures delivered at the 2013 AMS Short Course on Random Matrices, held January 6-7, 2013 in San Diego, California. Included are surveys by leading researchers in the field, written in introductory style, aiming to provide the reader a quick and intuitive overview of this fascinating and rapidly developing topic. These surveys contain many major recent developments, such as progress on universality conjectures, connections between random matrices and free probability, numerical algebra, combinatorics and high-dimensional geometry, together with several novel methods and a variety of open questions.

Author : James A. Mingo,Roland Speicher
Publisher : Springer
Page : 336 pages
File Size : 49,5 Mb
Release : 2017-06-24
Category : Mathematics
ISBN : 9781493969425

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Free Probability and Random Matrices by James A. Mingo,Roland Speicher Pdf

This volume opens the world of free probability to a wide variety of readers. From its roots in the theory of operator algebras, free probability has intertwined with non-crossing partitions, random matrices, applications in wireless communications, representation theory of large groups, quantum groups, the invariant subspace problem, large deviations, subfactors, and beyond. This book puts a special emphasis on the relation of free probability to random matrices, but also touches upon the operator algebraic, combinatorial, and analytic aspects of the theory. The book serves as a combination textbook/research monograph, with self-contained chapters, exercises scattered throughout the text, and coverage of important ongoing progress of the theory. It will appeal to graduate students and all mathematicians interested in random matrices and free probability from the point of view of operator algebras, combinatorics, analytic functions, or applications in engineering and statistical physics.

Author : John Harnad
Publisher : Springer Science & Business Media
Page : 526 pages
File Size : 55,8 Mb
Release : 2011-05-06
Category : Science
ISBN : 9781441995148

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Random Matrices, Random Processes and Integrable Systems by John Harnad Pdf

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.

Author : László Erdős,Horng-Tzer Yau
Publisher : American Mathematical Soc.
Page : 226 pages
File Size : 45,8 Mb
Release : 2017-08-30
Category : Random matrices
ISBN : 9781470436483

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A Dynamical Approach to Random Matrix Theory by László Erdős,Horng-Tzer Yau Pdf

A co-publication of the AMS and the Courant Institute of Mathematical Sciences at New York University This book is a concise and self-contained introduction of recent techniques to prove local spectral universality for large random matrices. Random matrix theory is a fast expanding research area, and this book mainly focuses on the methods that the authors participated in developing over the past few years. Many other interesting topics are not included, and neither are several new developments within the framework of these methods. The authors have chosen instead to present key concepts that they believe are the core of these methods and should be relevant for future applications. They keep technicalities to a minimum to make the book accessible to graduate students. With this in mind, they include in this book the basic notions and tools for high-dimensional analysis, such as large deviation, entropy, Dirichlet form, and the logarithmic Sobolev inequality. This manuscript has been developed and continuously improved over the last five years. The authors have taught this material in several regular graduate courses at Harvard, Munich, and Vienna, in addition to various summer schools and short courses. Titles in this series are co-published with the Courant Institute of Mathematical Sciences at New York University.

Author : Percy Deift,Dimitri Gioev
Publisher : American Mathematical Soc.
Page : 217 pages
File Size : 47,8 Mb
Release : 2009-01-01
Category : Mathematics
ISBN : 0821847376

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Random Matrix Theory by Percy Deift,Dimitri Gioev Pdf

This book features a unified derivation of the mathematical theory of the three classical types of invariant random matrix ensembles--orthogonal, unitary, and symplectic. The authors follow the approach of Tracy and Widom, but the exposition here contains a substantial amount of additional material, in particular, facts from functional analysis and the theory of Pfaffians. The main result in the book is a proof of universality for orthogonal and symplectic ensembles corresponding to generalized Gaussian type weights following the authors' prior work. New, quantitative error estimates are derived. The book is based in part on a graduate course given by the first author at the Courant Institute in fall 2005. Subsequently, the second author gave a modified version of this course at the University of Rochester in spring 2007. Anyone with some background in complex analysis, probability theory, and linear algebra and an interest in the mathematical foundations of random matrix theory will benefit from studying this valuable reference.

Author : AMS Short Course, Random Matrices
Publisher : Unknown
Page : 174 pages
File Size : 54,6 Mb
Release : 2014
Category : Electronic books
ISBN : 1470416603

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Modern Aspects of Random Matrix Theory by AMS Short Course, Random Matrices Pdf

Author : Richard A. Brualdi,Dragos Cvetkovic
Publisher : CRC Press
Page : 288 pages
File Size : 40,7 Mb
Release : 2008-08-06
Category : Mathematics
ISBN : 1420082248

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A Combinatorial Approach to Matrix Theory and Its Applications by Richard A. Brualdi,Dragos Cvetkovic Pdf

Unlike most elementary books on matrices, A Combinatorial Approach to Matrix Theory and Its Applications employs combinatorial and graph-theoretical tools to develop basic theorems of matrix theory, shedding new light on the subject by exploring the connections of these tools to matrices. After reviewing the basics of graph theory, elementary counting formulas, fields, and vector spaces, the book explains the algebra of matrices and uses the König digraph to carry out simple matrix operations. It then discusses matrix powers, provides a graph-theoretical definition of the determinant using the Coates digraph of a matrix, and presents a graph-theoretical interpretation of matrix inverses. The authors develop the elementary theory of solutions of systems of linear equations and show how to use the Coates digraph to solve a linear system. They also explore the eigenvalues, eigenvectors, and characteristic polynomial of a matrix; examine the important properties of nonnegative matrices that are part of the Perron–Frobenius theory; and study eigenvalue inclusion regions and sign-nonsingular matrices. The final chapter presents applications to electrical engineering, physics, and chemistry. Using combinatorial and graph-theoretical tools, this book enables a solid understanding of the fundamentals of matrix theory and its application to scientific areas.

Author : Bolian Liu,Hong-Jian Lai
Publisher : Springer Science & Business Media
Page : 317 pages
File Size : 41,7 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9781475731651

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Matrices in Combinatorics and Graph Theory by Bolian Liu,Hong-Jian Lai Pdf

Combinatorics and Matrix Theory have a symbiotic, or mutually beneficial, relationship. This relationship is discussed in my paper The symbiotic relationship of combinatorics and matrix theoryl where I attempted to justify this description. One could say that a more detailed justification was given in my book with H. J. Ryser entitled Combinatorial Matrix Theon? where an attempt was made to give a broad picture of the use of combinatorial ideas in matrix theory and the use of matrix theory in proving theorems which, at least on the surface, are combinatorial in nature. In the book by Liu and Lai, this picture is enlarged and expanded to include recent developments and contributions of Chinese mathematicians, many of which have not been readily available to those of us who are unfamiliar with Chinese journals. Necessarily, there is some overlap with the book Combinatorial Matrix Theory. Some of the additional topics include: spectra of graphs, eulerian graph problems, Shannon capacity, generalized inverses of Boolean matrices, matrix rearrangements, and matrix completions. A topic to which many Chinese mathematicians have made substantial contributions is the combinatorial analysis of powers of nonnegative matrices, and a large chapter is devoted to this topic. This book should be a valuable resource for mathematicians working in the area of combinatorial matrix theory. Richard A. Brualdi University of Wisconsin - Madison 1 Linear Alg. Applies., vols. 162-4, 1992, 65-105 2Camhridge University Press, 1991.