Probability Surveys

Determinantal Processes and Independence

J. Ben Hough, Manjunath Krishnapur, Yuval Peres, and Bálint Virág

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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.

Article information

Source
Probab. Surveys Volume 3 (2006), 206-229.

Dates
First available in Project Euclid: 5 May 2006

Permanent link to this document
http://projecteuclid.org/euclid.ps/1146832696

Digital Object Identifier
doi:10.1214/154957806000000078

Mathematical Reviews number (MathSciNet)
MR2216966

Zentralblatt MATH identifier
1189.60101

Citation

Hough, J. Ben; Krishnapur, Manjunath; Peres, Yuval; Virág, Bálint. Determinantal Processes and Independence. Probab. Surveys 3 (2006), 206--229. doi:10.1214/154957806000000078. http://projecteuclid.org/euclid.ps/1146832696.


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