Determinantal Processes and Independence

Details Determinantal Processes and Independence Determinantal Processes and Independence

2005-03-06

arxiv.org/abs/math/0503110v2

Probability Surveys 2006, Vol. 3, 206-229

Mathematics

Probability

Published at http://dx.doi.org/10.1214/154957806000000078 in the Probability Surveys (http://www.i-journals.org/ps/) by the In

Scientific paper

10.1214/154957806000000078

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 $L^2(D)$. 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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