**University Lecture Series**

Volume: 51;
2009;
154 pp;
Softcover

MSC: Primary 60; 30; 15;

Print ISBN: 978-0-8218-4373-4

Product Code: ULECT/51

List Price: $41.00

Individual Member Price: $32.80

**Electronic ISBN: 978-1-4704-1646-1
Product Code: ULECT/51.E**

List Price: $41.00

Individual Member Price: $32.80

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#### Supplemental Materials

# Zeros of Gaussian Analytic Functions and Determinantal Point Processes

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*J. Ben Hough; Manjunath Krishnapur; Yuval Peres; Bálint Virág*

The book examines in some depth two important classes of point processes, determinantal processes and “Gaussian zeros”, i.e., zeros of random analytic functions with Gaussian coefficients. These processes share a property of “point-repulsion”, where distinct points are less likely to fall close to each other than in processes, such as the Poisson process, that arise from independent sampling. Nevertheless, the treatment in the book emphasizes the use of independence: for random power series, the independence of coefficients is key; for determinantal processes, the number of points in a domain is a sum of independent indicators, and this yields a satisfying explanation of the central limit theorem (CLT) for this point count. Another unifying theme of the book is invariance of considered point processes under natural transformation groups.

The book strives for balance between general theory and concrete examples. On the one hand, it presents a primer on modern techniques on the interface of probability and analysis. On the other hand, a wealth of determinantal processes of intrinsic interest are analyzed; these arise from random spanning trees and eigenvalues of random matrices, as well as from special power series with determinantal zeros.

The material in the book formed the basis of a graduate course given at the IAS-Park City Summer School in 2007; the only background knowledge assumed can be acquired in first-year graduate courses in analysis and probability.

#### Table of Contents

# Table of Contents

## Zeros of Gaussian Analytic Functions and Determinantal Point Processes

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Preface vii8 free
- Introduction 112 free
- Gaussian analytic functions 1324 free
- Joint intensities 3546
- Determinantal point processes 4758
- The hyperbolic GAF 8394
- A determinantal zoo 99110
- Large deviations for zeros 119130
- Advanced topics: Dynamics and allocation to random zeros 135146
- Bibliography 149160
- Back Cover Back Cover1170

#### Readership

Graduate students and research mathematicians interested in random processes and their relations to complex analysis.