On random walk on growing graphs

TL;DR

This paper studies random walks on sequences of growing graphs, providing bounds and criteria for recurrence and transience based on graph properties, and addresses related universality and merging questions.

Contribution

It establishes transition probability bounds and recurrence/transience criteria for random walks on monotonically increasing graphs, advancing understanding of their long-term behavior.

Findings

Transition probability upper bounds for growing graphs

Sufficient transience criteria based on volume and Cheeger constant

Matching lower bounds and recurrence criteria in specialized cases

Abstract

Random walk on changing graphs is considered. For sequences of finite graphs increasing monotonically towards a limiting infinite graph, we establish transition probability upper bounds. It yields sufficient transience criteria for simple random walk on slowly growing graphs, upon knowing the volume and Cheeger constant of each graph. For much more specialized cases, we establish matching lower bounds, and deduce sufficient (weak) recurrence criteria. We also address recurrence directly in relation to a universality conjecture of [DHS]. We answer a related question of [SZ, Problem 1.8] about "inhomogeneous merging" in the negative.