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.

Metric spaces with dilations, and metric trees

Here is a gentle introduction to the theory of metric spaces with dilations (“rescaling maps”, so that one can define an appropriate notion of tangent space.) This appears on the arXiv today.

http://arxiv.org/abs/1007.2362

Also a couple of elegant-looking papers on metric trees and their embeddings into Banach spaces

http://arxiv.org/abs/1007.2207

http://arxiv.org/abs/1007.2208

Lots of interesting stuff on the arXiv today. I probably won’t be posting much for a few weeks as I have some personal business to take care of.