Once-reinforced random walk in high dimensions

TL;DR

This paper proves that once-reinforced random walks in dimensions six and higher are transient and behave diffusively, confirming a conjecture and employing novel capacity estimates and a game-theoretic approach.

Contribution

It establishes the transience and diffusive behavior of once-reinforced random walks in high dimensions, confirming a conjecture and introducing new analytical techniques.

Findings

Walks are transient for d≥6 with small reinforcement

Walks exhibit diffusive behavior and can be coupled with Brownian motion

Trajectory is nowhere heavy, as shown by capacity estimates

Abstract

We study the once-reinforced random walk on $\mathbb Z^d$, which is a self-interacting walk that has a higher probability to cross edges that were already visited. We prove that the walk is transient when $d\ge 6$ and when the reinforcement is small, establishing a conjecture of Sidoravicius in these dimensions. Moreover, in this case we prove that the walk behaves diffusively and can be coupled with Brownian motion. One of the main ideas in the proof is a certain capacity estimate which shows that the trajectory of the walk is nowhere heavy. We also use a game-theoretic-type ingredient that we call ``the demon" to force spatial independence in the process.