It was clear that the random-matrix ensembles possessed a deep mathematical structure. The Mehta-Gaudin calculation was the tip of an ice-berg, and I wanted to explore the whole ice-berg. I have continued to explore it for thirty years, at first continuously and later intermittently. Each time I return, it reveals new treasures.*
*From F. Dyson, Selected papers of Freeman Dyson with commentary, Collected Works, 5, American Mathematical Society, Providence, R.I.; International Press, Cambridge, Mass., 1996.
To Rebecca and Abby ― P.D. To my wife Diana Katsman and my mother Natalia Barkova ― D.G.
There has been a great upsurge of interest in random matrix theory (RMT) in recent years.
Therefor the proof of Theorem 6.51 is also complete, and so Theorem 6.7 and its corollaries, the main results of this text, are now finally proved.
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.
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