Remarks on random walks on graphs and the Floyd boundary

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, $p^{(n)} (v,w) \leq C \rho^n$, where $\rho \lt 1$ is the spectral radius, then for any Floyd function $f$ that satisfies $\sum^{\infty}_{n=1} nf(n) \lt \infty$, the Dirichlet problem with respect to the Floyd boundary is solvable.

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The author acknowledges the support of the Austrian Science Fund (FWF): W1230.

Received 30 April 2021

Received revised 17 September 2021

Accepted 2 November 2021

Published 16 May 2022