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arXiv · 2610.00090

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

Ahmed Bou-Rabee·Yuval Peres
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Abstract

In once-reinforced random walk, introduced by Davis (1990), every edge starts with weight one and is assigned weight $β\geq1$ after its first crossing. At each step the walker chooses an incident edge with probability proportional to its weight. For sufficiently large reinforcement $β$, 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 $β$.

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Ahmed Bou-Rabee, Yuval Peres. 2026-09-07. Once-reinforced random walk on $\mathbb{Z}^d$ has range exponent at least $d/(d+1)$. https://arxiv.org/abs/2610.00090

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