Harmonic functions on hyperbolic graphs

TL;DR

This paper investigates harmonic functions on hyperbolic graphs, establishing that non-tangential boundedness and convergence are almost everywhere equivalent, using geometric and probabilistic methods inspired by prior work on Riemannian manifolds and trees.

Contribution

It proves the almost everywhere equivalence of non-tangential boundedness and convergence for harmonic functions on hyperbolic graphs, extending known results to this setting.

Findings

Non-tangential boundedness and convergence are almost everywhere equivalent.

The proof combines geometric and probabilistic techniques.

Results extend classical theorems to hyperbolic graph structures.

Abstract

We consider admissible random walks on hyperbolic graphs. For a given harmonic function on such a graph, we prove that asymptotic properties of non-tangential boundedness and non-tangential convergence are almost everywhere equivalent. The proof is inspired by the works of F. Mouton in the cases of Riemannian manifolds of pinched negative curvature and infinite trees. It involves geometric and probabilitistic methods.