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Zeros of Gaussian analytic functions and determinantal point processes. (English) Zbl 1190.60038
University Lecture Series 51. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4373-4/pbk). ix, 154 p. (2009).
The book is devoted to classes of point processes such as determinantal processes and zeros of random analytic functions with complex Gaussian coefficients. The study of zeros of random polynomials goes back to Mark Kac who obtained the density of real zeros of random polynomials with real Gaussian distributed coefficients.
In the book under review, the authors derive expressions for the first intensity of the zero set of a general Gaussian analytic function. They show that the intensity of zeros of a Gaussian analytic function determines the distribution of the Gaussian analytic function. The authors find different expressions for the two-point intensity of zeros and specialize them to the canonical Gaussian analytic function. The book studies a number of examples of determinantal point processes that arise naturally in combinatorics and probability theory. The authors also study a large deviation estimate which is valid for arbitrary Gaussian analytic functions. The book strives for balance between general theory and concrete examples.

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
30B20 Random power series in one complex variable
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
60G15 Gaussian processes
15B52 Random matrices (algebraic aspects)