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Random Matrix Theory: Invariant Ensembles and Universality (Courant Lecture Notes)
This book features a unified derivation of the mathematical theory of the three classical types of invariant random matrix ensembles--orthogonal, unitary, and symplectic. The authors follow the approach of Tracy and Widom, but the exposition here contains a substantial amount of additional material, in particular, facts from functional analysis and the theory of Pfaffians. The main result in the book is a proof of universality for orthogonal and symplectic ensembles corresponding to generalized Gaussian type weights following the authors' prior work. New, quantitative error estimates are derived. The book is based in part on a graduate course given by the first author at the Courant Institute in fall 2005. Subsequently, the second author gave a modified version of this course at the University of Rochester in spring 2007. Anyone with some background in complex analysis, probability theory, and linear algebra and an interest in the mathematical foundations of random matrix theory will benefit from studying this valuable reference.
217 pages, Paperback
First published January 1, 2009
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January 9, 2015
Random matrix theory (RMT) has become quite a hot research area in applied mathematics over the past 25 years or so. This monograph is an outstanding addition to the literature on RMT, and is structured according to graduate-level courses offered by each author at their respective universities. This volume is an excellent complement/supplement to Terence Tao's "Topics in Random Matrix Theory" (surveyed by this reviewer approx. a month ago). Deift and Goiev's approach is more direct and pedagogical in the specific areas of RMT related to universality and invariants (see below), where Tao's text reveals some of the processes involved in evaluating and proving results in RMT. Both are at approximately the same level of mathematics, although for a student with no prior experience with RMT, the currect volume would be a better place to start, followed by the Tao book.
One of the things I appreciate about this monograph is the consistent reference to other areas of science with which RMT shares deep connections. One example is statistical mechanics, which deals with energy occupation levels in matter. Both RMT and statistical mechanics use something called a partition function, which is a function that typically appears in the denominator of expressions used to calculate mean energy in an ensemble of particles.
It should be noted that one of the central problems posed in RMT is the spectrum of eigenvalues of such matrices. If one considers a general random matrix - that is, a matrix whose elements are randomly generated from some general (perhaps even unspecified) probability distribution, and lacking even simple structure such as symmetry, then the answer to this question is rather intractable (though it continues to be a subject of research). Once one imposed some minimal structure on the matrix, e.g., the matrix conforms to construction rules for Markov transition matrices, or is orthogonal, Hermitian, or symplectic, and moreover that the underlying distribution function for the entries is something computationally tractable (this typically means Gaussian, or normal), then some fascinating and intriguing results emerge. Recalling that eigenvalues, and their associated eigenspaces, represent scale factors introduced by pre-multiplication of the matrix with vectors in the eigenspace, then the connection to things like energy occupancy levels begins to make sense. But the central question above remains. And what of the characteristics of this spectrum? If it is confined to the reals, or complex variables, is the spectrum dense in the underlying space? Are the components of the spectrum compact? What are the perturbational aspects of the eigenvalues and eigenspaces?
Three specific types of matrices are explored at length here: Gaussian Orthogonal Ensembles (GOE), Gaussian Unitary Ensembles (GUE), and Gaussian Symplectic Ensembles (GSE). Again, as mentioned above, Deift and Goiev's exposition both complements and supplements Tao's "exploratory" approach.
I recommend this book to any graduate-level mathematics student interested in RMT as an are of research, and to professional mathematicians as well. Typical background for this volume are advanced calculus, analysis of functions of one complex variable (the Cauchy residue formula appears frequently, as do notions of analytic and entire functions), advanced linear algebra, and measure theory.
One of the things I appreciate about this monograph is the consistent reference to other areas of science with which RMT shares deep connections. One example is statistical mechanics, which deals with energy occupation levels in matter. Both RMT and statistical mechanics use something called a partition function, which is a function that typically appears in the denominator of expressions used to calculate mean energy in an ensemble of particles.
It should be noted that one of the central problems posed in RMT is the spectrum of eigenvalues of such matrices. If one considers a general random matrix - that is, a matrix whose elements are randomly generated from some general (perhaps even unspecified) probability distribution, and lacking even simple structure such as symmetry, then the answer to this question is rather intractable (though it continues to be a subject of research). Once one imposed some minimal structure on the matrix, e.g., the matrix conforms to construction rules for Markov transition matrices, or is orthogonal, Hermitian, or symplectic, and moreover that the underlying distribution function for the entries is something computationally tractable (this typically means Gaussian, or normal), then some fascinating and intriguing results emerge. Recalling that eigenvalues, and their associated eigenspaces, represent scale factors introduced by pre-multiplication of the matrix with vectors in the eigenspace, then the connection to things like energy occupancy levels begins to make sense. But the central question above remains. And what of the characteristics of this spectrum? If it is confined to the reals, or complex variables, is the spectrum dense in the underlying space? Are the components of the spectrum compact? What are the perturbational aspects of the eigenvalues and eigenspaces?
Three specific types of matrices are explored at length here: Gaussian Orthogonal Ensembles (GOE), Gaussian Unitary Ensembles (GUE), and Gaussian Symplectic Ensembles (GSE). Again, as mentioned above, Deift and Goiev's exposition both complements and supplements Tao's "exploratory" approach.
I recommend this book to any graduate-level mathematics student interested in RMT as an are of research, and to professional mathematicians as well. Typical background for this volume are advanced calculus, analysis of functions of one complex variable (the Cauchy residue formula appears frequently, as do notions of analytic and entire functions), advanced linear algebra, and measure theory.
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