Determinantal probability measures

Determinantal point processes have arisen in diverse settings in recent years and have been investigated intensively. We study basic combinatorial and probabilistic aspects in the discrete case. Our main results concern relationships with matroids, stochastic domination, negative association, completeness for infinite matroids, tail triviality, and a method for extension of results from orthogonal projections to positive contractions. We also present several new avenues for further investigation, involving Hilbert spaces, combinatorics, homology, and group representations, among other areas.Comment: 50 pp; added reference to revision. Revised introduction and made other small change

Distance covariance in metric spaces

We extend the theory of distance (Brownian) covariance from Euclidean spaces, where it was introduced by Sz\'{e}kely, Rizzo and Bakirov, to general metric spaces. We show that for testing independence, it is necessary and sufficient that the metric space be of strong negative type. In particular, we show that this holds for separable Hilbert spaces, which answers a question of Kosorok. Instead of the manipulations of Fourier transforms used in the original work, we use elementary inequalities for metric spaces and embeddings in Hilbert spaces.Comment: Published in at http://dx.doi.org/10.1214/12-AOP803 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Hyperbolic Space Has Strong Negative Type

It is known that hyperbolic spaces have strict negative type, a condition on the distances of any finite subset of points. We show that they have strong negative type, a condition on every probability distribution of points (with integrable distance to a fixed point). This implies that the function of expected distances to points determines the probability measure uniquely. It also implies that the distance covariance test for stochastic independence, introduced by Sz\'ekely, Rizzo and Bakirov, is consistent against all alternatives in hyperbolic spaces. We prove this by showing an analogue of the Cram\'er-Wold device.Comment: 6 p

Factors of IID on Trees

Classical ergodic theory for integer-group actions uses entropy as a complete invariant for isomorphism of IID (independent, identically distributed) processes (a.k.a. product measures). This theory holds for amenable groups as well. Despite recent spectacular progress of Bowen, the situation for non-amenable groups, including free groups, is still largely mysterious. We present some illustrative results and open questions on free groups, which are particularly interesting in combinatorics, statistical physics, and probability. Our results include bounds on minimum and maximum bisection for random cubic graphs that improve on all past bounds.Comment: 18 pages, 1 figur