Random walk in slowly changing environments

TL;DR

This paper establishes that bounded random walks in slowly changing environments on graphs retain the recurrence or transience properties of the initial environment, providing a broad criterion applicable to various graph structures.

Contribution

It introduces a general condition showing that slowly varying edge-weights do not alter the fundamental recurrence or transience nature of the walk.

Findings

Bounded RWCEs inherit initial recurrence/transience properties

Condition applies to any locally finite, connected graph

Results extend to self-interacting random walks with history-dependent weights

Abstract

A Random Walk in Changing Environment (RWCE) is a weighted random walk on a locally finite, connected graph $G$ with random, time-dependent edge-weights. This includes self-interacting random walks, where the edge-weights depend on the history of the process. In general, even the basic question of recurrence or transience for RWCEs is difficult, especially when the underlying graph contains cycles. In this note, we derive a condition for recurrence or transience that is too restrictive for classical RWCEs but instead works for any graph $G.$ Namely, we show that any bounded RWCE on $G$ with "slowly" changing edge-weights inherits the recurrence or transience of the initial weighted graph.