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A Dynamical Approach to Random Matrix Theory

László Erdős and Horng-Tzer Yau
Publisher: 
American Mathematical Society
Publication Date: 
2017
Number of Pages: 
226
Format: 
Paperback
Series: 
Courant Lecture Notes 28
Price: 
43.00
ISBN: 
9781470436483
Category: 
Monograph
[Reviewed by
William J. Satzer
, on
12/18/2017
]

Random matrices first entered the mathematical literature around 1928, when Wishart studied the distribution of sample covariance matrices. Later, Wigner considered random matrices in nuclear physics when he noticed in experimental data that the spacing between lines in the spectrum of heavy metals resembled gaps between eigenvalues of random matrices, and that the statistics appeared largely independent of the material. Since then research on random matrix theory has grown considerably in many directions including number theory, physics, statistics, neuroscience, control theory, image analysis and image compression.

The current book focuses on a set of techniques used to resolve a conjecture of local spectral universality for large random matrices. The conjecture says roughly that the local eigenvalue statistics for large random matrices with independent entries are universal in the sense that they do not depend on the particular distribution of the matrix elements. The conjecture is true for the important special cases of Gaussian Orthogonal and Unitary Ensembles and was proved by explicitly calculating the gap distribution functions and local correlation functions of the eigenvalues. The broader question has not been amenable to direct computation.

The authors’ goal is to provide a self-contained introduction to a new approach to local spectral universality that resolves the conjecture. For expository purposes, they work with a smaller set of matrices (generalized Wigner matrices) and an averaged-energy version of universality. The idea is to focus on key ideas and concepts and develop the most important techniques. “Self-contained” here means something like “no need to have specific background with random matrix theory”, but the reader would still need a strong background in analysis at the graduate level and comparable comfort with probabilistic arguments.

The work is in in three parts. The first expands Wigner’s original semicircle law to a local law for generalized Wigner matrices. The second proves local universality for Gaussian divisible ensembles, and the third expands local spectral universality from Gaussian divisible ensembles to all Wigner ensembles. It’s the second part that is critical and novel, and the part that merits the “dynamics” of the book’s title.

This is a monograph aimed more at experts or those wanting to develop expertise in this area. For readers who might want a broader introduction to the topic, Mehta’s Random Matrices is probably a good starting point. This book is not the place to go for a reader with only casual interest in the subject.

 

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Bill Satzer ([email protected]) was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.