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 orthe domain of localization of the eigenvalues are varied. The resulting differential equations determining the partition function and correlation functionsare, 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 to 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. Correlation functions between eigenvalues of random matrices also have close similarities to those in integrable quantum spin systems and many body models, with determinantal expressions of very similar form appearing in both. There are also remarkable connections to further probabilistic problems such as random words, tilings and partitions, as well as related growth processes. Provides an in-depth examination of random matrices with applications overa vast variety of domains, including multivariate statistics, random growth models, and many othersApplies the theory of integrable systems, a source of powerful analytic methods, to the solution of fundamental problems in random systems and processesFeatures an interdisciplinary approach that sheds new light on a dynamic topic of current researchExplains and develops the phenomenon of 'universality,' in particular, the occurrence of the Tracy-Widom distribution for eigenvalues at the 'edge of the spectrum,' in the longest increasing subsequence of a random permutation and a variety of critical phenomena in the double scaling limit INDICE: Introduction by John Harnad Part I Random Matrices, Randopm Processes and Integrable Models. Chapter 1 Random and Integrable Models in Mathematics and Physics by Pierre van Moerbeke. Chapter 2 Integrable Systems, Random Matrices, and Random Processes by Mark Adler Part II Random Matrices and Applications. Chapter 3 Integral Operators in Random Matrix Theory by Harold Widom.Chapter 4 Lectures on Random Matrix Models by Pavel M. Bleher. Chapter 5 Large N Asymptotics in Random Matrices by Alexander R. Its. Chapter 6 Formal Matrix Integrals and Combinatorics of Maps by B. Eynard. Chapter 7 Application of Random Matrix Theory to Multivariate Statistics by Momar Dieng and Craig A. Tracy
- ISBN: 978-1-4419-9513-1
- Editorial: Springer New York
- Encuadernacion: Cartoné
- Páginas: 515
- Fecha Publicación: 28/06/2011
- Nº Volúmenes: 1
- Idioma: Inglés