Almost Sure Recurrence of the Simple Random Walk Path

TL;DR

This paper proves that the set of vertices and edges visited by a simple random walk on any graph forms a subgraph that is almost surely recurrent, meaning the walk returns infinitely often to visited vertices.

Contribution

It establishes the almost sure recurrence of the random walk path as a subgraph on any graph, a novel general recurrence result.

Findings

Random walk path is almost surely recurrent on any graph.

Visited subgraph contains infinitely many returns.

Universal recurrence property for simple random walks.

Abstract

It is shown that the path of a simple random walk on any graph, consisting of all vertices visited and edges crossed by the walk, is almost surely a recurrent subgraph.