Once reinforced random walk on $\mathbb{Z}\times \Gamma$

TL;DR

This paper proves recurrence of once reinforced random walk on infinite cylinders for large reinforcement, establishes a shape theorem for visited sites, and analyzes fluctuations, using new estimates on subgraph visitation times.

Contribution

It demonstrates recurrence of the process on $ ext{Z} imes ext{finite graph}$ for large reinforcement and introduces a shape theorem with polynomial fluctuation bounds.

Findings

Recurrence established for large reinforcement parameters.

Shape theorem describing the visited set.

Polynomial order fluctuations around the shape.

Abstract

We revisit an unpublished paper of Vervoort (2002) on the once reinforced random walk, and prove that this process is recurrent on any graph of the form $\mathbb{Z}\times \Gamma$, with $\Gamma$ a finite graph, for sufficiently large reinforcement parameter. We also obtain a shape theorem for the set of visited sites, and show that the fluctuations around this shape are of polynomial order. The proof involves sharp general estimates on the time spent on subgraphs of the ambiant graph which might be of independent interest.