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Determinantal processes and independence. (English) Zbl 1189.60101

Summary: 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.

MSC:

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60F05 Central limit and other weak theorems
82B31 Stochastic methods applied to problems in equilibrium statistical mechanics
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