Asymptotics of Random Matrices and Related Models: The Uses of Dyson-Schwinger Equations
Title | Asymptotics of Random Matrices and Related Models: The Uses of Dyson-Schwinger Equations PDF eBook |
Author | Alice Guionnet |
Publisher | American Mathematical Soc. |
Pages | 143 |
Release | 2019-04-29 |
Genre | Green's functions |
ISBN | 1470450275 |
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.
Random Matrices, Random Processes and Integrable Systems
Title | Random Matrices, Random Processes and Integrable Systems PDF eBook |
Author | John Harnad |
Publisher | Springer Science & Business Media |
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.
Lectures on Random Lozenge Tilings
Title | Lectures on Random Lozenge Tilings PDF eBook |
Author | Vadim Gorin |
Publisher | Cambridge University Press |
Pages | 261 |
Release | 2021-09-09 |
Genre | Language Arts & Disciplines |
ISBN | 1108843964 |
This is the first book dedicated to reviewing the mathematics of random tilings of large domains on the plane.
Asymptotic Expansion of a Partition Function Related to the Sinh-model
Title | Asymptotic Expansion of a Partition Function Related to the Sinh-model PDF eBook |
Author | Gaëtan Borot |
Publisher | Springer |
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.
Large Random Matrices: Lectures on Macroscopic Asymptotics
Title | Large Random Matrices: Lectures on Macroscopic Asymptotics PDF eBook |
Author | Alice Guionnet |
Publisher | Springer |
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.
Random Matrices and the Six-Vertex Model
Title | Random Matrices and the Six-Vertex Model PDF eBook |
Author | Pavel Bleher |
Publisher | American Mathematical Soc. |
Pages | 237 |
Release | 2013-12-04 |
Genre | Mathematics |
ISBN | 1470409615 |
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.
Random Matrices and Their Applications
Title | Random Matrices and Their Applications PDF eBook |
Author | Joel E. Cohen |
Publisher | American Mathematical Soc. |
Pages | 380 |
Release | 1986-12-31 |
Genre | Mathematics |
ISBN | 9780821853986 |
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.