Download or read book Random Matrix Theory Interacting Particle Systems and Integrable Systems written by Percy Deift and published by Cambridge University Press. This book was released on 2014-12-15 with total page 539 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume includes review articles and research contributions on long-standing questions on universalities of Wigner matrices and beta-ensembles.
Download or read book Random Matrices Random Processes and Integrable Systems written by John Harnad and published by Springer Science & Business Media. This book was released on 2011-05-06 with total page 536 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book On Several Problems in Random Matrix Theory and Statistical Mechanics written by Yuanyuan Xu and published by . This book was released on 2018 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: 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).
Download or read book Integrable Systems and Random Matrices written by Jinho Baik and published by American Mathematical Soc.. This book was released on 2008 with total page 448 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Particle Systems Random Media and Large Deviations written by Richard Durrett and published by American Mathematical Soc.. This book was released on 1985 with total page 394 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Random Matrix Theory And Its Applications Multivariate Statistics And Wireless Communications written by Zhidong Bai and published by World Scientific. This book was released on 2009-07-27 with total page 176 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Random Matrix Models and Their Applications written by Pavel Bleher and published by Cambridge University Press. This book was released on 2001-06-04 with total page 454 pages. Available in PDF, EPUB and Kindle. Book excerpt: Expository articles on random matrix theory emphasizing the exchange of ideas between the physical and mathematical communities.
Download or read book A First Course in Random Matrix Theory written by Marc Potters and published by Cambridge University Press. This book was released on 2020-12-03 with total page 371 pages. Available in PDF, EPUB and Kindle. Book excerpt: An intuitive, up-to-date introduction to random matrix theory and free calculus, with real world illustrations and Big Data applications.
Download or read book Random Matrices written by Alexei Borodin and published by American Mathematical Soc.. This book was released on 2019-10-30 with total page 498 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Nonlinear Dispersive Partial Differential Equations and Inverse Scattering written by Peter D. Miller and published by Springer Nature. This book was released on 2019-11-14 with total page 528 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains lectures and invited papers from the Focus Program on "Nonlinear Dispersive Partial Differential Equations and Inverse Scattering" held at the Fields Institute from July 31-August 18, 2017. The conference brought together researchers in completely integrable systems and PDE with the goal of advancing the understanding of qualitative and long-time behavior in dispersive nonlinear equations. The program included Percy Deift’s Coxeter lectures, which appear in this volume together with tutorial lectures given during the first week of the focus program. The research papers collected here include new results on the focusing nonlinear Schrödinger (NLS) equation, the massive Thirring model, and the Benjamin-Bona-Mahoney equation as dispersive PDE in one space dimension, as well as the Kadomtsev-Petviashvili II equation, the Zakharov-Kuznetsov equation, and the Gross-Pitaevskii equation as dispersive PDE in two space dimensions. The Focus Program coincided with the fiftieth anniversary of the discovery by Gardner, Greene, Kruskal and Miura that the Korteweg-de Vries (KdV) equation could be integrated by exploiting a remarkable connection between KdV and the spectral theory of Schrodinger's equation in one space dimension. This led to the discovery of a number of completely integrable models of dispersive wave propagation, including the cubic NLS equation, and the derivative NLS equation in one space dimension and the Davey-Stewartson, Kadomtsev-Petviashvili and Novikov-Veselov equations in two space dimensions. These models have been extensively studied and, in some cases, the inverse scattering theory has been put on rigorous footing. It has been used as a powerful analytical tool to study global well-posedness and elucidate asymptotic behavior of the solutions, including dispersion, soliton resolution, and semiclassical limits.
Download or read book A Dynamical Approach to Random Matrix Theory written by László Erdős and published by American Mathematical Soc.. This book was released on 2017-08-30 with total page 239 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Free Probability and Random Matrices written by James A. Mingo and published by Springer. This book was released on 2017-06-24 with total page 343 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book The Random Matrix Theory of Quantum Many particle Systems written by and published by . This book was released on 2011 with total page 33 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Asymptotics of Random Matrices and Related Models The Uses of Dyson Schwinger Equations written by Alice Guionnet and published by American Mathematical Soc.. This book was released on 2019-04-29 with total page 143 pages. Available in PDF, EPUB and Kindle. Book excerpt: Probability theory is based on the notion of independence. The celebrated law of large numbers and the central limit theorem describe the asymptotics of the sum of independent variables. However, there are many models of strongly correlated random variables: for instance, the eigenvalues of random matrices or the tiles in random tilings. Classical tools of probability theory are useless to study such models. These lecture notes describe a general strategy to study the fluctuations of strongly interacting random variables. This strategy is based on the asymptotic analysis of Dyson-Schwinger (or loop) equations: the author will show how these equations are derived, how to obtain the concentration of measure estimates required to study these equations asymptotically, and how to deduce from this analysis the global fluctuations of the model. The author will apply this strategy in different settings: eigenvalues of random matrices, matrix models with one or several cuts, random tilings, and several matrices models.
Download or read book An Introduction to Integrable Techniques for One Dimensional Quantum Systems written by Fabio Franchini and published by Springer. This book was released on 2017-05-25 with total page 186 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces the reader to basic notions of integrable techniques for one-dimensional quantum systems. In a pedagogical way, a few examples of exactly solvable models are worked out to go from the coordinate approach to the Algebraic Bethe Ansatz, with some discussion on the finite temperature thermodynamics. The aim is to provide the instruments to approach more advanced books or to allow for a critical reading of research articles and the extraction of useful information from them. We describe the solution of the anisotropic XY spin chain; of the Lieb-Liniger model of bosons with contact interaction at zero and finite temperature; and of the XXZ spin chain, first in the coordinate and then in the algebraic approach. To establish the connection between the latter and the solution of two dimensional classical models, we also introduce and solve the 6-vertex model. Finally, the low energy physics of these integrable models is mapped into the corresponding conformal field theory. Through its style and the choice of topics, this book tries to touch all fundamental ideas behind integrability and is meant for students and researchers interested either in an introduction to later delve in the advance aspects of Bethe Ansatz or in an overview of the topic for broadening their culture.
Download or read book Tau Functions and their Applications written by John Harnad and published by Cambridge University Press. This book was released on 2021-02-04 with total page 549 pages. Available in PDF, EPUB and Kindle. Book excerpt: Tau functions are a central tool in the modern theory of integrable systems. This volume provides a thorough introduction, starting from the basics and extending to recent research results. It covers a wide range of applications, including generating functions for solutions of integrable hierarchies, correlation functions in the spectral theory of random matrices and combinatorial generating functions for enumerative geometrical and topological invariants. A self-contained summary of more advanced topics needed to understand the material is provided, as are solutions and hints for the various exercises and problems that are included throughout the text to enrich the subject matter and engage the reader. Building on knowledge of standard topics in undergraduate mathematics and basic concepts and methods of classical and quantum mechanics, this monograph is ideal for graduate students and researchers who wish to become acquainted with the full range of applications of the theory of tau functions.
Download or read book The Random Matrix Theory of the Classical Compact Groups written by Elizabeth S. Meckes and published by Cambridge University Press. This book was released on 2019-08-01 with total page 225 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the first book to provide a comprehensive overview of foundational results and recent progress in the study of random matrices from the classical compact groups, drawing on the subject's deep connections to geometry, analysis, algebra, physics, and statistics. The book sets a foundation with an introduction to the groups themselves and six different constructions of Haar measure. Classical and recent results are then presented in a digested, accessible form, including the following: results on the joint distributions of the entries; an extensive treatment of eigenvalue distributions, including the Weyl integration formula, moment formulae, and limit theorems and large deviations for the spectral measures; concentration of measure with applications both within random matrix theory and in high dimensional geometry; and results on characteristic polynomials with connections to the Riemann zeta function. This book will be a useful reference for researchers and an accessible introduction for students in related fields.
