Self-interacting random walks

TL;DR

This paper studies self-interacting random walks in multi-dimensional space, establishing conditions for their transience or recurrence, and provides a complete classification in three dimensions.

Contribution

It offers new criteria for transience and constructs examples of recurrence, especially clarifying the behavior in three-dimensional cases.

Findings

In dimension 3, all two-measure walks are transient.

Existence of recurrent walks generated by three measures in dimension 3.

Conditions for transience of self-interacting random walks.

Abstract

Let $\mu_1,... \mu_k$ be $d$-dimensional probability measures in $\R^d$ with mean 0. At each step we choose one of the measures based on the history of the process and take a step according to that measure. We give conditions for transience of such processes and also construct examples of recurrent processes of this type. In particular, in dimension 3 we give the complete picture: every walk generated by two measures is transient and there exists a recurrent walk generated by three measures.