Random Matrices, Frobenius Eigenvalues, and Monodromy

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The main topic of this book is the deep relation between the spacings between zeros of zeta and $L$-functions and spacings between eigenvalues of random elements of large compact classical groups. This relation, the Montgomery-Odlyzko law, is shown to hold for wide classes of zeta and $L$-functions over finite fields. The book draws on and gives accessible accounts of many disparate areas of mathematics, from algebraic geometry, moduli spaces, monodromy, equidistribution, and the Weil conjectures, to probability theory on the compact classical groups in the limit as their dimension goes to infinity and related techniques from orthogonal polynomials and Fredholm determinants.

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Editorial Reviews

Mathematicians from Princeton University focus on the Montgomery- Odlyzko law, the deep relation between the spacings between zeros of zeta and L-functions and spacings between eigenvalues of random elements of large compact classical groups. Finds the law to hold for wide classes of zeta and L-functions over finite fields. Of interest to research mathematicians and graduate students studying such areas as varieties over finite and local fields, zeta-functions, limit theorems, and the structure of families. Annotation c. by Book News, Inc., Portland, Or.
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Product Details

Table of Contents

Introduction 1
Ch. 1 Statements of the Main Results 17
Ch. 2 Reformulation of the Main Results 35
Ch. 3 Reduction Steps in Proving the Main Theorems 73
Ch. 4 Test Functions 101
Ch. 5 Haar Measure 107
Ch. 6 Tail Estimates 141
Ch. 7 Large N Limits and Fredholm Determinants 197
Ch. 8 Several Variables 245
Ch. 9 Equidistribution 267
Ch. 10 Monodromy of Families of Curves 293
Ch. 11 Monodromy of Some Other Families 323
Ch. 12 GUE Discrepancies in Various Families 351
Ch. 13 Distribution of Low-lying Frobenius Eigenvalues in Various Families 361
App Densities 373
App: Graphs 411
References 417
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