Year: 2026
Type: preprint
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 $β$.
Source: arXiv (Cornell University)
Authors: Bou-Rabee, Ahmed, Peres, Yuval
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7216079110