Preface

Summary

In the spring of 1999, MSRI hosted a very successful and influential onesemester program on random matrix theory (RMT) and its applications. At the workshops during the semester, there was a sense of excitement as brand new and very recent results were reported. The goal of the 2010 Program has been to showcase the many remarkable developments that have taken place since 1999 and to spur further developments in RMT and related areas of interacting particle systems (IPS) and integrable systems (IS) as well as to highlight various applications of RMT.

One of the outputs of the 1999 program was volume 40 in the MSRI Publications series, entitled “Random matrix models and their applications”. Looking back on this publication today, it is clear that this volume gave a representative snapshot of topics that were occupying the attention of researchers in the field then. Moreover, the papers—consisting of a mix of research articles and reviews— provide a conveniently bundled resource for researchers in the field to this day.

Since 1999 random matrix theory has captured the imagine of a whole new generation of researchers, and through a collective effort some outstanding questions have been settled, and new highly promising research areas initiated. One example of the former is work on universality questions forWigner matrices, where the task is to show that for large dimension a symmetric matrix with independent entries of mean zero and standard deviation 1 has the same statistical properties as in the case of standard Gaussian entries. Another is universality questions for β-ensembles, where one wants to show that the statistical properties are independent of the one-body potential. New research areas include the KPZ equation and related growth processes, which has led to the precise experimental realization of some random matrix distributions, and also to quite spectacular theoretical advances relating to a rigorous understanding of the replica trick via so called Mcdonald processes; analytic predictions of the -generalization of the so-called Dyson constant in the asymptotic expansion of spacing distributions inβ-ensembles; and stochastic differential equations and PDEs for eigenvalue distributions in the case of a low rank perturbation leading to eigenvalue separation.