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| Author | : Virgil Pierce |
| Publisher | : |
| Total Pages | : 0 |
| Release | : 2004 |
| Genre | : |
| ISBN | : |
| Author | : Gaëtan Borot |
| Publisher | : Springer |
| Total Pages | : 233 |
| Release | : 2016-12-08 |
| Genre | : Science |
| ISBN | : 3319333798 |
This book elaborates on the asymptotic behaviour, when N is large, of certain N-dimensional integrals which typically occur in random matrices, or in 1+1 dimensional quantum integrable models solvable by the quantum separation of variables. The introduction presents the underpinning motivations for this problem, a historical overview, and a summary of the strategy, which is applicable in greater generality. The core aims at proving an expansion up to o(1) for the logarithm of the partition function of the sinh-model. This is achieved by a combination of potential theory and large deviation theory so as to grasp the leading asymptotics described by an equilibrium measure, the Riemann-Hilbert approach to truncated Wiener-Hopf in order to analyse the equilibrium measure, the Schwinger-Dyson equations and the boostrap method to finally obtain an expansion of correlation functions and the one of the partition function. This book is addressed to researchers working in random matrices, statistical physics or integrable systems, or interested in recent developments of asymptotic analysis in those fields.
| Author | : Alice Guionnet |
| Publisher | : Springer |
| Total Pages | : 296 |
| Release | : 2009-04-20 |
| Genre | : Mathematics |
| ISBN | : 3540698973 |
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 | : John Harnad |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 536 |
| Release | : 2011-05-06 |
| Genre | : 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 | : Zhonggen Su |
| Publisher | : World Scientific |
| Total Pages | : 284 |
| Release | : 2015-04-20 |
| Genre | : Mathematics |
| ISBN | : 9814612243 |
This book is aimed at graduate students and researchers who are interested in the probability limit theory of random matrices and random partitions. It mainly consists of three parts. Part I is a brief review of classical central limit theorems for sums of independent random variables, martingale differences sequences and Markov chains, etc. These classical theorems are frequently used in the study of random matrices and random partitions. Part II concentrates on the asymptotic distribution theory of Circular Unitary Ensemble and Gaussian Unitary Ensemble, which are prototypes of random matrix theory. It turns out that the classical central limit theorems and methods are applicable in describing asymptotic distributions of various eigenvalue statistics. This is attributed to the nice algebraic structures of models. This part also studies the Circular β Ensembles and Hermitian β Ensembles. Part III is devoted to the study of random uniform and Plancherel partitions. There is a surprising similarity between random matrices and random integer partitions from the viewpoint of asymptotic distribution theory, though it is difficult to find any direct link between the two finite models. A remarkable point is the conditioning argument in each model. Through enlarging the probability space, we run into independent geometric random variables as well as determinantal point processes with discrete Bessel kernels.This book treats only second-order normal fluctuations for primary random variables from two classes of special random models. It is written in a clear, concise and pedagogical way. It may be read as an introductory text to further study probability theory of general random matrices, random partitions and even random point processes.
| Author | : Giacomo Livan |
| Publisher | : Springer |
| Total Pages | : 122 |
| Release | : 2018-01-16 |
| Genre | : Science |
| ISBN | : 3319708856 |
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 | : Jun Yan (Researcher in random matrix theory) |
| Publisher | : |
| Total Pages | : |
| Release | : 2020 |
| Genre | : |
| ISBN | : |
Random matrix theory has a long history. It was first introduced in mathematical statistics by John Wishart in 1928, and it gained attention during the 1950s due to work by Eugene Wigner studying the distribution of nuclear energy levels. A large number of physicists and mathematicians have been fascinated by random matrix theory, and after decades of study, it has matured into a field with applications in many branches of physics and mathematics. Nowadays, the subject is still very much alive with new and exciting research. Much of my PhD work has revolved around the study of random matrix theory. This dissertation gives a tour of my work on asymptotic theory of large random matrices and its applications in statistics, probability, and the theory of orthogonal polynomials, respectively.
| Author | : James A. Mingo |
| Publisher | : Springer |
| Total Pages | : 343 |
| Release | : 2017-06-24 |
| Genre | : Mathematics |
| ISBN | : 1493969420 |
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 | : Édouard Brezin |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 519 |
| Release | : 2006-07-03 |
| Genre | : Science |
| ISBN | : 140204531X |
Random matrices are widely and successfully used in physics for almost 60-70 years, beginning with the works of Dyson and Wigner. Although it is an old subject, it is constantly developing into new areas of physics and mathematics. It constitutes now a part of the general culture of a theoretical physicist. Mathematical methods inspired by random matrix theory become more powerful, sophisticated and enjoy rapidly growing applications in physics. Recent examples include the calculation of universal correlations in the mesoscopic system, new applications in disordered and quantum chaotic systems, in combinatorial and growth models, as well as the recent breakthrough, due to the matrix models, in two dimensional gravity and string theory and the non-abelian gauge theories. The book consists of the lectures of the leading specialists and covers rather systematically many of these topics. It can be useful to the specialists in various subjects using random matrices, from PhD students to confirmed scientists.
| Author | : Roman Vershynin |
| Publisher | : Cambridge University Press |
| Total Pages | : 299 |
| Release | : 2018-09-27 |
| Genre | : Business & Economics |
| ISBN | : 1108415199 |
An integrated package of powerful probabilistic tools and key applications in modern mathematical data science.
