## Probability Surveys

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

### Determinantal Processes and Independence

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

#### 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
*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.

#### 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.