Recurrent and (strongly) resolvable graphs

TL;DR

This paper introduces a novel method for analyzing recurrence and harmonic functions on infinite weighted graphs using capacity and intrinsic metrics, linking boundary polar sets with path null sets.

Contribution

It presents a new approach based on capacity and intrinsic metrics, connecting boundary polar sets with path null sets through diverging finite energy functions.

Findings

Established a link between polar sets and null path sets

Developed a new capacity-based framework for recurrence analysis

Provided tools for studying harmonic functions on infinite graphs

Abstract

We develop a new approach to recurrence and the existence of non-constant harmonic functions on infinite weighted graphs. The approach is based on the capacity of subsets of metric boundaries with respect to intrinsic metrics. The main tool is a connection between polar sets in such boundaries and null sets of paths. This connection relies on suitably diverging functions of finite energy.