Phase transition for the Once-reinforced random walk on $\mathbb{Z}^d$-like trees

TL;DR

This paper establishes the existence of a phase transition in the recurrence and transience behavior of the Once-reinforced random walk on certain trees, characterized by an explicit critical reinforcement parameter.

Contribution

It provides the first known examples of phase transition for the Once-reinforced random walk on trees with bounded degree.

Findings

Existence of a critical parameter a_0 for recurrence/transience transition.

Almost sure recurrence when a > a_0.

Almost sure transience when a < a_0.

Abstract

In this short paper, we consider the Once-reinforced random walk with reinforcement parameter $a$ on trees with bounded degree which are transient for the simple random walk. On each of these trees, we prove that there exists an explicit critical parameter $a_0$ such that the Once-reinforced random walk is almost surely recurrent if $a>a_0$ and almost surely transient if $a<a_0$. This provides the first examples of phase transition for the Once-reinforced random walk.