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 $\beta \ge 1$ 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 $d\ge 2$ and for every reinforcement parameter $\beta$.

arXiv comment15 pages, 2 figures. Lean 4 formalization: https://github.com/nitromannitol/ORRW-Lower-Bound