Showing papers by "Hilary K. Finucane published in 2013"
TL;DR: In this paper, initial steps towards understanding which finitely generated groups are almost surely generated as a semigroup by the path of a random walk on the group are presented, and initial steps are presented towards understanding how groups are generated as semigroups.
Abstract: Initial steps are presented towards understanding which finitely generated groups are almost surely generated as a semigroup by the path of a random walk on the group.
TL;DR: In this article, the authors considered the problem of finding the largest number of Voronoi cells that must be infinite, given sufficiently (but finitely) many sites which are sufficiently far from each other.
Abstract: In this paper, we consider the Voronoi decompositions of an arbitrary infinite vertex-transitive graph G. In particular, we are interested in the following question: what is the largest number of Voronoi cells that must be infinite, given sufficiently (but finitely) many Voronoi sites which are sufficiently far from each other? We call this number the survival number s(G). The survival number of a graph has an alternative characterization in terms of the number of balls of radius r-1 required to cover a sphere of radius r. The survival number is not a quasi-isometry invariant, but it remains open whether finiteness of s(G) is. We show that all vertex-transitive graphs with polynomial growth have finite s(G); vertex-transitive graphs with infinitely many ends have infinite s(G); the lamplighter graph LL(Z), which has exponential growth, has finite s(G); and the lamplighter graph LL(Z2), which is Liouville, has infinite s(G).