Random Walk in Changing Environment

TL;DR

This paper introduces and analyzes the concept of Random Walks in Changing Environments, where the underlying graph or weights change over time, exploring properties like recurrence and transience with various criteria and examples.

Contribution

It formalizes the notion of RW in dynamic environments, providing criteria for recurrence/transience and examples, including a novel transient walk on b62 with changing conductances.

Findings

Provided criteria for recurrence and transience in changing environments

Constructed an example of a transient walk on b62 with changing conductances

Conjectured limitations on behavior when weights are predetermined

Abstract

In this paper we introduce the notion of Random Walk in Changing Environment - a random walk in which each step is performed in a different graph on the same set of vertices, or more generally, a weighted random walk on the same vertex and edge sets but with different (possibly 0) weights in each step. This is a very wide class of RW, which includes some well known types of RW as special cases (e.g. reinforced RW, true SAW). We define and explore various possible properties of such walks, and provide criteria for recurrence and transience when the underlying graph is $\mathbb{N}$ or a tree. We provide an example of such a process on $\mathbb{Z}^2$ where conductances can only change from $1$ to $2$ (once for each edge) but nevertheless the walk is transient, and conjecture that such behaviour cannot happen when the weights are chosen in advance, that is, do not depend on the location of the RW.