Remarks on random walks on graphs and the Floyd boundary

TL;DR

This paper establishes conditions under which random walks on graphs converge to the Floyd boundary and solves the Dirichlet problem using Floyd functions, linking graph properties with boundary behavior.

Contribution

It introduces a Floyd boundary convergence result for uniformly irreducible random walks with bounded range and solves the Dirichlet problem under spectral radius decay conditions.

Findings

Random walks with bounded range converge to Floyd boundary.

Under spectral radius decay, the Dirichlet problem is solvable for certain Floyd functions.

Provides a framework connecting random walk behavior with boundary theory.

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