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Processes on Unimodular Random Networks
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.Expand
Probability on Trees and Networks
Starting around the late 1950s, several research communities began relating the geometry of graphs to stochastic processes on these graphs. This book, twenty years in the making, ties togetherExpand
A Simple Path to Biggins’ Martingale Convergence for Branching Random Walk
We give a simple non-analytic proof of Biggins’ theorem on martingale convergence for branching random walks.
Conceptual proofs of L log L criteria for mean behavior of branching processes
The Kesten-Stigum theorem is a fundamental criterion for the rate of growth of a supercritical branching process, showing that an L log L condition is decisive. In critical and subcritical cases,Expand
Random Walks and Percolation on Trees
There is a way to define an average number of branches per vertex for an arbitrary infinite locally finite tree. It equals the exponential of the Hausdorff dimension of the boundary in an appropriateExpand
Uniform spanning forests
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)Expand
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. OurExpand
Indistinguishability of Percolation Clusters
We show that when percolation produces infinitely many infinite clusters on a Cayley graph, one cannot distinguish the clusters from each other by any invariantly defined property. This implies thatExpand
Group-invariant Percolation on Graphs
Abstract. Let G be a closed group of automorphisms of a graph X. We relate geometric properties of G and X, such as amenability and unimodularity, to properties of G-invariant percolation processesExpand
Random walk in a random environment and rst-passage percolation on trees
A delay line refresh memory stores the bits to be displayed on a visual display means such as a television receiver. A shift register in the feedback loop applies the stored bits back to the inputExpand