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Author : Percy Deift
Publisher : Cambridge University Press
Page : 539 pages
File Size : 48,92 MB
Release : 2014-12-15
Category : Language Arts & Disciplines
ISBN : 1107079926
This volume includes review articles and research contributions on long-standing questions on universalities of Wigner matrices and beta-ensembles.
Author : John Harnad
Publisher : Springer Science & Business Media
Page : 536 pages
File Size : 42,99 MB
Release : 2011-05-06
Category : Science
ISBN : 1441995145
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 : Yuanyuan Xu
Publisher :
Page : pages
File Size : 47,66 MB
Release : 2018
Category :
ISBN : 9780438290754
Random Matrix Theory(RMT) is a fast developing area of modern Mathematics with deep connections to Probability, Statistical Mechanics, Quantum Theory, Number Theory, Statistics, and Integrable Systems. In the first part of my dissertation, I consider an interacting particle system on the unit circle with stronger repulsion than that of the Circular beta Ensemble in RMT and prove the Gaussian approximation of the distribution of the particles. In addition, the Central Limit Theorem(CLT) for the linear statistics of the particles is obtained as a corollary. In the second part of the dissertation, I consider the orthogonal group SO(2n) with the Haar measure and prove the CLT for the linear eigenvalue statistics in the mesoscopic regime where the test function depends on n. The results can be generalized to other classic compact groups, such as SO(2n+1) and Sp(n).
Author : Jinho Baik
Publisher : American Mathematical Soc.
Page : 448 pages
File Size : 12,26 MB
Release : 2008
Category : Mathematics
ISBN : 0821842404
This volume contains the proceedings of a conference held at the Courant Institute in 2006 to celebrate the 60th birthday of Percy A. Deift. The program reflected the wide-ranging contributions of Professor Deift to analysis with emphasis on recent developments in Random Matrix Theory and integrable systems. The articles in this volume present a broad view on the state of the art in these fields. Topics on random matrices include the distributions and stochastic processes associated with local eigenvalue statistics, as well as their appearance in combinatorial models such as TASEP, last passage percolation and tilings. The contributions in integrable systems mostly deal with focusing NLS, the Camassa-Holm equation and the Toda lattice. A number of papers are devoted to techniques that are used in both fields. These techniques are related to orthogonal polynomials, operator determinants, special functions, Riemann-Hilbert problems, direct and inverse spectral theory. Of special interest is the article of Percy Deift in which he discusses some open problems of Random Matrix Theory and the theory of integrable systems.
Author : Richard Durrett
Publisher : American Mathematical Soc.
Page : 394 pages
File Size : 30,81 MB
Release : 1985
Category : Mathematics
ISBN : 0821850423
Covers the proceedings of the 1984 AMS Summer Research Conference. This work provides a summary of results from some of the areas in probability theory; interacting particle systems, percolation, random media (bulk properties and hydrodynamics), the Ising model and large deviations.
Author : Zhidong Bai
Publisher : World Scientific
Page : 176 pages
File Size : 48,32 MB
Release : 2009-07-27
Category : Mathematics
ISBN : 9814467995
Random matrix theory has a long history, beginning in the first instance in multivariate statistics. It was used by Wigner to supply explanations for the important regularity features of the apparently random dispositions of the energy levels of heavy nuclei. The subject was further deeply developed under the important leadership of Dyson, Gaudin and Mehta, and other mathematical physicists.In the early 1990s, random matrix theory witnessed applications in string theory and deep connections with operator theory, and the integrable systems were established by Tracy and Widom. More recently, the subject has seen applications in such diverse areas as large dimensional data analysis and wireless communications.This volume contains chapters written by the leading participants in the field which will serve as a valuable introduction into this very exciting area of research.
Author : Pavel Bleher
Publisher : Cambridge University Press
Page : 454 pages
File Size : 43,16 MB
Release : 2001-06-04
Category : Mathematics
ISBN : 9780521802093
Expository articles on random matrix theory emphasizing the exchange of ideas between the physical and mathematical communities.
Author : Marc Potters
Publisher : Cambridge University Press
Page : 371 pages
File Size : 39,21 MB
Release : 2020-12-03
Category : Computers
ISBN : 1108488080
An intuitive, up-to-date introduction to random matrix theory and free calculus, with real world illustrations and Big Data applications.
Author : László Erdős
Publisher : American Mathematical Soc.
Page : 239 pages
File Size : 30,66 MB
Release : 2017-08-30
Category : Mathematics
ISBN : 1470436485
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 : Alexei Borodin
Publisher : American Mathematical Soc.
Page : 498 pages
File Size : 17,43 MB
Release : 2019-10-30
Category : Education
ISBN : 1470452804
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.