Remarks on random walks on graphs and the Floyd boundary

Abstract

We show that for a uniformly irreducible random walk on a graph, with bounded range, there is a Floyd function for which the random walk converges to its corresponding Floyd boundary. Moreover if we add the assumptions, p(n)(v,w) ≤ Cρ, where ρ < 1 is the spectral radius, then for any Floyd function f that satisfies ∑∞ n=1 nf(n) < ∞, the Dirichlet problem with respect to the Floyd boundary is solvable.