Asymptotics Of Random Matrices And Related Models The Uses Of Dyson Schwinger Equations

Author : Alice Guionnet
Publisher : American Mathematical Soc.
Page : 143 pages
File Size : 45,8 Mb
Release : 2019-04-29
Category : Green's functions
ISBN : 9781470450274

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Asymptotics of Random Matrices and Related Models: The Uses of Dyson-Schwinger Equations by Alice Guionnet Pdf

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.

Author : John Harnad
Publisher : Springer Science & Business Media
Page : 536 pages
File Size : 48,6 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 : Vadim Gorin
Publisher : Cambridge University Press
Page : 261 pages
File Size : 54,9 Mb
Release : 2021-09-09
Category : Language Arts & Disciplines
ISBN : 9781108843966

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Lectures on Random Lozenge Tilings by Vadim Gorin Pdf

This is the first book dedicated to reviewing the mathematics of random tilings of large domains on the plane.

Author : Alice Guionnet
Publisher : Springer
Page : 296 pages
File Size : 53,5 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 : Pavel Bleher, Karl Liechty
Publisher : American Mathematical Soc.
Page : 237 pages
File Size : 50,9 Mb
Release : 2013-12-04
Category : Mathematics
ISBN : 9781470409616

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Random Matrices and the Six-Vertex Model by Pavel Bleher, Karl Liechty Pdf

This book provides a detailed description of the Riemann-Hilbert approach (RH approach) to the asymptotic analysis of both continuous and discrete orthogonal polynomials, and applications to random matrix models as well as to the six-vertex model. The RH approach was an important ingredient in the proofs of universality in unitary matrix models. This book gives an introduction to the unitary matrix models and discusses bulk and edge universality. The six-vertex model is an exactly solvable two-dimensional model in statistical physics, and thanks to the Izergin-Korepin formula for the model with domain wall boundary conditions, its partition function matches that of a unitary matrix model with nonpolynomial interaction. The authors introduce in this book the six-vertex model and include a proof of the Izergin-Korepin formula. Using the RH approach, they explicitly calculate the leading and subleading terms in the thermodynamic asymptotic behavior of the partition function of the six-vertex model with domain wall boundary conditions in all the three phases: disordered, ferroelectric, and antiferroelectric. Titles in this series are co-published with the Centre de Recherches Mathématiques.

Author : László Erdős,Horng-Tzer Yau
Publisher : American Mathematical Soc.
Page : 226 pages
File Size : 52,5 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 : Pavel Bleher,Alexander Its
Publisher : Cambridge University Press
Page : 454 pages
File Size : 45,7 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 : Gaëtan Borot,Alice Guionnet,Karol K. Kozlowski
Publisher : Springer
Page : 222 pages
File Size : 41,5 Mb
Release : 2016-12-08
Category : Science
ISBN : 9783319333793

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Asymptotic Expansion of a Partition Function Related to the Sinh-model by Gaëtan Borot,Alice Guionnet,Karol K. Kozlowski Pdf

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 : Édouard Brezin
Publisher : Springer Science & Business Media
Page : 528 pages
File Size : 40,5 Mb
Release : 2006-03-03
Category : Mathematics
ISBN : 1402045301

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Applications of Random Matrices in Physics by Édouard Brezin Pdf

Proceedings of the NATO Advanced Study Institute on Applications of Random Matrices in Physics, Les Houches, France, 6-25 June 2004

Author : Grégory Schehr,Alexander Altland,Yan V. Fyodorov,Neil O'Connell,Leticia F. Cugliandolo
Publisher : Oxford University Press
Page : 432 pages
File Size : 44,6 Mb
Release : 2017-08-15
Category : Science
ISBN : 9780192517869

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Stochastic Processes and Random Matrices by Grégory Schehr,Alexander Altland,Yan V. Fyodorov,Neil O'Connell,Leticia F. Cugliandolo Pdf

The field of stochastic processes and Random Matrix Theory (RMT) has been a rapidly evolving subject during the last fifteen years. The continuous development and discovery of new tools, connections and ideas have led to an avalanche of new results. These breakthroughs have been made possible thanks, to a large extent, to the recent development of various new techniques in RMT. Matrix models have been playing an important role in theoretical physics for a long time and they are currently also a very active domain of research in mathematics. An emblematic example of these recent advances concerns the theory of growth phenomena in the Kardar-Parisi-Zhang (KPZ) universality class where the joint efforts of physicists and mathematicians during the last twenty years have unveiled the beautiful connections between this fundamental problem of statistical mechanics and the theory of random matrices, namely the fluctuations of the largest eigenvalue of certain ensembles of random matrices. This text not only covers this topic in detail but also presents more recent developments that have emerged from these discoveries, for instance in the context of low dimensional heat transport (on the physics side) or integrable probability (on the mathematical side).

Author : Alexei Borodin,Ivan Corwin,Alice Guionnet
Publisher : American Mathematical Soc.
Page : 498 pages
File Size : 49,5 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 : Joel E. Cohen,Harry Kesten,Charles Michael Newman,American Mathematical Society
Publisher : American Mathematical Soc.
Page : 380 pages
File Size : 48,5 Mb
Release : 1986-12-31
Category : Mathematics
ISBN : 0821853988

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Random Matrices and Their Applications by Joel E. Cohen,Harry Kesten,Charles Michael Newman,American Mathematical Society Pdf

These twenty-six expository papers on random matrices and products of random matrices survey the major results of the last thirty years. They reflect both theoretical and applied concerns in fields as diverse as computer science, probability theory, mathematical physics, and population biology. Many of the articles are tutorial, consisting of examples, sketches of proofs, and interpretations of results. They address a wide audience of mathematicians and scientists who have an elementary knowledge of probability theory and linear algebra, but not necessarily any prior exposure to this specialized area. More advanced articles, aimed at specialists in allied areas, survey current research with references to the original literature. The book's major topics include the computation and behavior under perturbation of Lyapunov exponents and the spectral theory of large random matrices. The applications to mathematical and physical sciences under consideration include computer image generation, card shuffling, and other random walks on groups, Markov chains in random environments, the random Schroedinger equations and random waves in random media. Most of the papers were originally presented at an AMS-IMS-SIAM Joint Summer Research Conference held at Bowdoin College in June, 1984. Of special note are the papers by Kotani on random Schroedinger equations, Yin and Bai on spectra for large random matrices, and Newman on the relations between the Lyapunov and eigenvalue spectra.

Author : Zhonggen Su
Publisher : World Scientific
Page : 284 pages
File Size : 41,5 Mb
Release : 2015-04-20
Category : Mathematics
ISBN : 9789814612241

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Random Matrices And Random Partitions: Normal Convergence by Zhonggen Su Pdf

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 : Leonid Andreevich Pastur,Mariya Shcherbina
Publisher : American Mathematical Soc.
Page : 650 pages
File Size : 41,7 Mb
Release : 2011
Category : Mathematics
ISBN : 9780821852859

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Eigenvalue Distribution of Large Random Matrices by Leonid Andreevich Pastur,Mariya Shcherbina Pdf

Random matrix theory is a wide and growing field with a variety of concepts, results, and techniques and a vast range of applications in mathematics and the related sciences. The book, written by well-known experts, offers beginners a fairly balanced collection of basic facts and methods (Part 1 on classical ensembles) and presents experts with an exposition of recent advances in the subject (Parts 2 and 3 on invariant ensembles and ensembles with independent entries). The text includes many of the authors' results and methods on several main aspects of the theory, thus allowing them to present a unique and personal perspective on the subject and to cover many topics using a unified approach essentially based on the Stieltjes transform and orthogonal polynomials. The exposition is supplemented by numerous comments, remarks, and problems. This results in a book that presents a detailed and self-contained treatment of the basic random matrix ensembles and asymptotic regimes. This book will be an important reference for researchers in a variety of areas of mathematics and mathematical physics. Various chapters of the book can be used for graduate courses; the main prerequisite is a basic knowledge of calculus, linear algebra, and probability theory.

Author : Anonim
Publisher : Unknown
Page : 1572 pages
File Size : 55,9 Mb
Release : 1992
Category : Aeronautics
ISBN : UIUC:30112078757413

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Scientific and Technical Aerospace Reports by Anonim Pdf