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2006 Determinantal Processes and Independence
J. Ben Hough, Manjunath Krishnapur, Yuval Peres, Bálint Virág
Probab. Surveys 3: 206-229 (2006). DOI: 10.1214/154957806000000078

Abstract

We give a probabilistic introduction to determinantal and permanental point processes. Determinantal processes arise in physics (fermions, eigenvalues of random matrices) and in combinatorics (nonintersecting paths, random spanning trees). They have the striking property that the number of points in a region D is a sum of independent Bernoulli random variables, with parameters which are eigenvalues of the relevant operator on L2(D). Moreover, any determinantal process can be represented as a mixture of determinantal projection processes. We give a simple explanation for these known facts, and establish analogous representations for permanental processes, with geometric variables replacing the Bernoulli variables. These representations lead to simple proofs of existence criteria and central limit theorems, and unify known results on the distribution of absolute values in certain processes with radially symmetric distributions.

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J. Ben Hough. Manjunath Krishnapur. Yuval Peres. Bálint Virág. "Determinantal Processes and Independence." Probab. Surveys 3 206 - 229, 2006. https://doi.org/10.1214/154957806000000078

Information

Published: 2006
First available in Project Euclid: 5 May 2006

zbMATH: 1189.60101
MathSciNet: MR2216966
Digital Object Identifier: 10.1214/154957806000000078

Rights: Copyright © 2006 The Institute of Mathematical Statistics and the Bernoulli Society

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