Determinantal point processes

I went to a talk by Russ Lyons of Indiana this morning.  The subject, an intriguing one to me, was the relationship between costs and \(\ell^2\) Betti numbers in the context of discrete groups.  In the course of the discussion, though, the notion of determinantal process showed up, and I wanted to get acquainted with that.  Russ has an article about the generalities of this idea (referenced below) and there is also a nice blog post by Terry Tao.

Let \(E\) be a set (at most countable for now).  We’re interested in probability measures on the set of all subsets of \(E\) – in the jargon of probability theory, such a measure is called a point process on \(E\).  For example, one such measure is given by fixing a probability \(p\) and then determining independently, with probability \(p\), whether each \(e\in E\) is or is not a member of a random subset.  This is called the Bernoulli process with the given probability.

Suppose a point process is given, and let \(\mathfrak S\) denote a random subset of \(E\) for that process. We are interested in the finite marginals of the process: these are the probabilities

\[ {\mathbb P}( e_1,\ldots,e_k \in {\mathfrak S}) \]

for finite subsets \( \{e_1,\ldots,e_k\} \) of \(E\).  (In terms of measure theory, these are the measures of the cylinder subsets of the power set of \(E\).)  For example, the marginals of the Bernoulli process are just \(p^k\).

Definition: We say that the process is determinantal if there is a symmetric positive kernel \(K\) on \(E\) such that the finite marginals are given by determinants, as follows,

\[ {\mathbb P}( e_1,\ldots,e_k \in {\mathfrak S}) = \det \bigl( K(e_i,e_j)_{i,j=1,\ldots,k} \bigr) \]

For example, the Bernoulli process is determinantal.  The corresponding \(K\) just has entries \(p\) down the diagonal, and zeroes elsewhere.

Lyons establishes a correspondence between determinantal processes on \(E\) and positive contractions on the Hilbert space \(\ell^2(E)\).   Taking contractions related to the combinatorial Laplacian of a graph then produces processes related to random spanning trees.

“Determinantal Processes.”   Accessed October 24. http://terrytao.wordpress.com/2009/08/23/determinantal-processes/.

Lyons, Russell. 2003. “Determinantal Probability Measures.” Publications Mathématiques De l’Institut Des Hautes Études Scientifiques 98 (1): 167–212. doi:10.1007/s10240-003-0016-0.

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