We investigate unimodular random networks. Our motivations include their characterization via reversibility of an associated random walk and their similarities to unimodular quasi-transitive graphs.â€¦ (More)

A love and respect of trees has been characteristic of mankind since the beginning of human evolution. Instinctively, we understood the importance of trees to our lives before we were able to ascribeâ€¦ (More)

We study uniform spanning forest measures on infinite graphs, which are weak limits of uniform spanning tree measures from finite subgraphs. These limits can be taken with free (FSF) or wired (WSF)â€¦ (More)

We give a simple non-analytic proof of Bigginsâ€™ theorem on martingale convergence for branching random walks. Let L := {Xi} L i=1 be a random L-tuple of real numbers, where L is also random and canâ€¦ (More)

We show that the transience or recurrence of a random walk in certain random environments on an arbitrary infinite locally finite tree is determined by the branching number of the tree, which is aâ€¦ (More)

Determinantal point processes have arisen in diverse settings in recent years and have been investigated intensively. We initiate a detailed study of the discrete analogue, the most prominent exampleâ€¦ (More)

We show that independent percolation on any Cayley graph of a nonamenable group has no innnite components at the critical parameter. This result was obtained in Benjamini, Lyons, Peres, and Schrammâ€¦ (More)

We give new formulas for the asymptotics of the number of spanning trees of a large graph. A special case answers a question of McKay [Europ. J. Combin. 4 149â€“160] for regular graphs. The generalâ€¦ (More)

We survey known results about phase transitions in various models of statistical physics when the underlying space is a nonamenable graph. Most attention is devoted to transitive graphs and trees.