On the range of random walk on graphs satisfying a uniform condition

TL;DR

This paper investigates the behavior of the range of random walks on graphs satisfying a specific uniform potential-theoretic condition, establishing weak laws and demonstrating fluctuations in mean behavior.

Contribution

It extends understanding of random walk ranges to a broad class of graphs, including non-regular ones, and shows the optimality of weak laws under the condition.

Findings

Weak laws of R_n established from above and below

Existence of graphs with fluctuating mean R_n/n

Weak laws are optimal under the uniform condition

Abstract

We consider the range of random walks up to time n, R_n, on graphs satisfying a uniform condition. This condition is characterized by potential theory. Not only all vertex transitive graphs but also many non-regular graphs satisfy the condition. We show certain weak laws of R_n from above and below. We also show that there is a graph such that it satisfies the condition and a sequence of the mean of R_n/n fluctuates. By noting the construction of the graph, we see that under the condition, the weak laws are best in a sense.