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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 together… Expand

A Simple Path to Biggins’ Martingale Convergence for Branching Random Walk

- R. Lyons
- Mathematics
- 23 March 1998

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

- R. Lyons, R. Pemantle, Y. Peres
- Mathematics
- 1 July 1995

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

- R. Lyons
- Mathematics
- 1 July 1990

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 appropriate… Expand

Uniform spanning forests

- I. Benjamini, R. Lyons, Y. Peres, O. Schramm
- Mathematics
- 1 February 2001

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

- R. Lyons
- Mathematics
- 26 April 2002

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… Expand

Indistinguishability of Percolation Clusters

- R. Lyons, O. Schramm
- Mathematics, Physics
- 29 November 1998

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 that… Expand

Group-invariant Percolation on Graphs

- I. Benjamini, R. Lyons, Y. Peres, O. Schramm
- Mathematics
- 1 March 1999

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 processes… Expand

Random walk in a random environment and rst-passage percolation on trees

- R. Lyons, R. Pemantle
- Computer Science, Mathematics
- 2 April 2004

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 input… Expand

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