Random Matrices, Random Processes and Integrable Systems (eBook)

John Harnad (Herausgeber)

eBook Download: PDF

2011 | 2011
XVIII, 526 Seiten
Springer New York (Verlag)
978-1-4419-9514-8 (ISBN)

Infos

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.

---

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 "e;Dyson processes"e;, 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.

Inhalt

Introduction by John HarnadPart I Random Matrices, Random Processes and Integrable ModelsChapter 1 Random and Integrable Models in Mathematics and Physics by Pierre van MoerbekeChapter 2 Integrable Systems, Random Matrices, and Random Processes by Mark Adler Part II Random Matrices and ApplicationsChapter 3 Integral Operators in Random Matrix Theory by Harold WidomChapter 4 Lectures on Random Matrix Models by Pavel M. BleherChapter 5 Large N Asymptotics in Random Matrices by Alexander R. ItsChapter 6 Formal Matrix Integrals and Combinatorics of Maps by B. EynardChapter 7 Application of Random Matrix Theory to Multivariate Statistics by Momar Dieng and Craig A. Tracy

Produkt-Details

• Erscheint lt. Verlag: 6.5.2011
• Reihe/Serie: CRM Series in Mathematical Physics
• Reihe/Serie: CRM Series in Mathematical Physics
• Zusatzinfo: XVIII, 526 p.
• Verlagsort: New York
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Allgemeines / Lexika
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Astronomie / Astrophysik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Theoretische Physik
• Themenwelt: Technik
• Schlagworte: Integrable Systems • nonlinear steepest descent • random growth models • random matrices • random processes • random sequences • Riemann-Hibert method
• ISBN-10: 1-4419-9514-5 / 1441995145
• ISBN-13: 978-1-4419-9514-8 / 9781441995148