PREFACE The purpose of this volume is to describe some recent and not so recent discoveries about random matrices and their applications to a wide audience of scientists and pure and applied mathematicians. The authors were requested to provide exposition, not detailed reports of new research to give simple, concrete and vivid examples of significant phenomena, rather than abstract but inaccessible theorems that might cover all possibilities. The papers provide heuristic interpretations of results, sketch the ideas of proofs, and point to or comment on the original reports of research. The papers draw on or describe applications in the fields of probability theory, functional analysis, group theory, mathematical physics, statistics, computer science, population biology, and number theory. All papers were refereed for expository and technical quality. The editors are grateful for the co-operative efforts of the authors and the referees. The papers in this volume fall into two broad groups: those concerned with products of random matrices and those in which the multiplication of random matrices plays no role. Interest in random matrices per se has a fairly long tradition in statistics and in physics (e.g. Mehta, 1967) [citations are given in the bibliography at the end of the book). Interest in products of random matrices can be dated from the first published paper on the topic by Bellman (1956). The results about the asymptotic behavior of products of random matrices discovered by Furstenberg and Kesten (1960) led to much of the interest in products of random matrices over the last twenty-five years, as many of the papers on products of random matrices in this volume demonstrate. Most of these papers were presented at a meeting on random matrices and their applications, organized as an AMS-IMS-SIAM Joint Summer Research Conference in the Mathematical Sciences at Bowdoin College, Brunswick, Maine, 17-23 June 1984. This may have been the first, but was certainly not the only, conference devoted to random matrices and their applications. Simultaneously with the Bowdoin meeting, a meeting on products of random matrices was held in Toulouse, France. In November 1984 another meeting on Lyapounov exponents was held in Bremen, West Germany. xiii

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