A combinatorial result with applications to self-interacting random walks

TL;DR

This paper presents combinatorial results related to arrow collections that help analyze self-interacting random walks, leading to reproofs of known results and new insights into multi-excited and random environment walks.

Contribution

It introduces combinatorial tools applicable to self-interacting random walks, providing new results and simplified proofs for recurrence and transience in various models.

Findings

Reproved known transience and recurrence results

Established new results for multi-excited random walks

Extended analysis to random walks in random environments

Abstract

We give a series of combinatorial results that can be obtained from any two collections (both indexed by $\Z\times \N$) of left and right pointing arrows that satisfy some natural relationship. When applied to certain self-interacting random walk couplings, these allow us to reprove some known transience and recurrence results for some simple models. We also obtain new results for one-dimensional multi-excited random walks and for random walks in random environments in all dimensions.