Branching Processes, and Random-Cluster Measures on Trees

Random-cluster measures on infinite regular trees are studied in conjunction with a general type of `boundary condition', namely an equivalence relation on the set of infinite paths of the tree. The uniqueness and non-uniqueness of random-cluster measures are explored for certain classes of equivalence relations. In proving uniqueness, the following problem concerning branching processes is encountered and answered. Consider bond percolation on the family-tree $T$ of a branching process. What is the probability that every infinite path of $T$, beginning at its root, contains some vertex which is itself the root of an infinite open sub-tree

Unimodularity of Invariant Random Subgroups

An invariant random subgroup $H \leq G$ is a random closed subgroup whose law is invariant to conjugation by all elements of $G$. When $G$ is locally compact and second countable, we show that for every invariant random subgroup $H \leq G$ there almost surely exists an invariant measure on $G/H$. Equivalently, the modular function of $H$ is almost surely equal to the modular function of $G$, restricted to $H$. We use this result to construct invariant measures on orbit equivalence relations of measure preserving actions. Additionally, we prove a mass transport principle for discrete or compact invariant random subgroups.Comment: 23 pages, one figur

Sur les espaces mesures singuliers II - Etude spectrale

We are mainly interested here in Kazhdan's property T for measured equivalence relations. Among our main results are characterizations of strong ergodicity and Kazhdan's property in terms of the spectra of diffusion operators, associated to random walks and hilbertian representations of the underlying equivalence relation. The analog spectral characterization of property T for countable groups was proved recently by Gromov \cite{Gromov03} (and Ghys \cite{Ghys03}). Our proof put together the tools developed in the group case and further crucial technical steps from the study of amenable equivalence relations in \cite{CFW81}. As an application we show how \.Zuk's "$\lambda_1 >1/2$" criterion for property T can be adapted to measured equivalence relations.Comment: french, english summar