Log-Gases and Random Matrices (LMS-34)

by Peter J. Forrester

Overview

Random matrix theory, both as an application and as a theory, has evolved rapidly over the past fifteen years. Log-Gases and Random Matrices gives a comprehensive account of these developments, emphasizing log-gases as a physical picture and heuristic, as well as covering topics such as beta ensembles and Jack polynomials.


Peter Forrester presents an encyclopedic development of log-gases and random matrices viewed as examples of integrable or exactly solvable systems. Forrester develops not only the application and theory of Gaussian and circular ensembles of classical random matrix theory, but also of the Laguerre and Jacobi ensembles, and their beta extensions. Prominence is given to the computation of a multitude of Jacobians; determinantal point processes and orthogonal polynomials of one variable; the Selberg integral, Jack polynomials, and generalized hypergeometric functions; Painlevé transcendents; macroscopic electrostatistics and asymptotic formulas; nonintersecting paths and models in statistical mechanics; and applications of random matrix theory. This is the first textbook development of both nonsymmetric and symmetric Jack polynomial theory, as well as the connection between Selberg integral theory and beta ensembles. The author provides hundreds of guided exercises and linked topics, making Log-Gases and Random Matrices an indispensable reference work, as well as a learning resource for all students and researchers in the field.

Product Details

ISBN-13:	9780691128290
Publisher:	Princeton University Press
Publication date:	07/21/2010
Series:	London Mathematical Society Monographs
Pages:	808
Product dimensions:	8.20(w) x 10.00(h) x 1.60(d)
Age Range:	18 Years

About the Author

Peter J. Forrester is professor of mathematics at the University of Melbourne.

Table of Contents

• FrontMatter, pg. i
• Preface, pg. v
• Contents, pg. xi
• Chapter One. Gaussian Matrix Ensembles, pg. 1
• Chapter Two. Circular Ensembles, pg. 53
• Chapter Three. Laguerre And Jacobi Ensembles, pg. 85
• Chapter Four. The Selberg Integral, pg. 133
• Chapter Five. Correlation functions at β = 2, pg. 186
• Chapter Six. Correlation Functions At β= 1 And 4, pg. 236
• Chapter Seven. Scaled limits at β = 1, 2 and 4, pg. 283
• Chapter Eight. Eigenvalue probabilities – Painlevé systems approach, pg. 328
• Chapter Nine. Eigenvalue probabilities— Fredholm determinant approach, pg. 380
• Chapter Ten. Lattice paths and growth models, pg. 440
• Chapter Eleven. The Calogero–Sutherland model, pg. 505
• Chapter Twelve. Jack polynomials, pg. 543
• Chapter Thirteen. Correlations for general β, pg. 592
• Chapter Fourteen. Fluctuation formulas and universal behavior of correlations, pg. 658
• Chapter Fifteen. The two-dimensional one-component plasma, pg. 701
• Bibliography, pg. 765
• Index, pg. 785

What People are Saying About This

Boris Khoruzhenko

This self-contained treatment starts from the basics and leads to the 'high end' of the subject. Forrester often gives new derivations of old results that beginners will find helpful, and the coverage of comprehensive topics will be useful to practitioners in the field.
— Boris Khoruzhenko, Queen Mary, University of London

From the Publisher

"Encyclopedic in scope, this book achieves an excellent balance between the theoretical and physical approaches to the subject. It coherently leads the reader from first-principle definitions, through a combination of physical and mathematical arguments, to the full derivation of many fundamental results. The vast amount of material and impeccable choice of topics make it an invaluable reference."—Eduardo Dueñez, University of Texas, San Antonio

"This self-contained treatment starts from the basics and leads to the 'high end' of the subject. Forrester often gives new derivations of old results that beginners will find helpful, and the coverage of comprehensive topics will be useful to practitioners in the field."—Boris Khoruzhenko, Queen Mary, University of London

Eduardo Duenez

Encyclopedic in scope, this book achieves an excellent balance between the theoretical and physical approaches to the subject. It coherently leads the reader from first-principle definitions, through a combination of physical and mathematical arguments, to the full derivation of many fundamental results. The vast amount of material and impeccable choice of topics make it an invaluable reference.
— Eduardo Duenez, University of Texas, San Antonio

Editorial Reviews

"Log-Gases and Random Matrices is an excellent book. It is bound to become an instant classic and the standard reference to a large body of contemporary random matrix theory. It is a well-written tour through a vast landscape. The contemporary literature is extensively referenced and incorporated in the text, and the material is presented from several perspectives. Forrester has achieved the pedagogical equivalent of Dyson's 'Threefold Way' by writing an advanced monograph appealing equally to physicists, mathematicians, and statisticians."—-Steven Joel Miller and Eduardo Duenez, Mathematical Reviews

From the Publisher

Log-Gases and Random Matrices is an excellent book. It is bound to become an instant classic and the standard reference to a large body of contemporary random matrix theory. It is a well-written tour through a vast landscape. The contemporary literature is extensively referenced and incorporated in the text, and the material is presented from several perspectives. Forrester has achieved the pedagogical equivalent of Dyson's 'Threefold Way' by writing an advanced monograph appealing equally to physicists, mathematicians, and statisticians.

Mathematical Reviews - Steven Joel Miller and Eduardo Duenez

Log-Gases and Random Matrices is an excellent book. It is bound to become an instant classic and the standard reference to a large body of contemporary random matrix theory. It is a well-written tour through a vast landscape. The contemporary literature is extensively referenced and incorporated in the text, and the material is presented from several perspectives. Forrester has achieved the pedagogical equivalent of Dyson's 'Threefold Way' by writing an advanced monograph appealing equally to physicists, mathematicians, and statisticians.
— Steven Joel Miller and Eduardo Duenez

Mathematical Reviews