Once-reinforced random walk on $\mathbb{Z}^d$ has range exponent at least $d/(d+1)$

Abstract

In once-reinforced random walk, introduced by Davis (1990), every edge starts with weight one and is assigned weight $\beta\geq1$ after its first crossing. At each step the walker chooses an incident edge with probability proportional to its weight. For sufficiently large reinforcement $\beta$, the expected range in the first $n$ steps was predicted by Ordemann et al. (2000) and independently by Beffara (2011) to have order $n^{d/(d+1)}$ on $\mathbb{Z}^d$. We prove the corresponding lower bound in every dimension $d\geq2$ and for every reinforcement parameter $\beta$.