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Once-reinforced random walk on ℤd 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 β≥1 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 nd/(d+1) on ℤd. We prove the corresponding lower bound in every dimension d≥2 and for every reinforcement parameter β.
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BibTeX
@misc{bourabee2026oncereinforced,
title = {{Once-reinforced random walk on $\mathbbZ^d$ has range exponent at least $d/(d+1)$}},
author = {Ahmed Bou-Rabee and Yuval Peres},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00090},
eprint = {2610.00090},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00090},
}
RIS
TY - UNPB
TI - Once-reinforced random walk on ℤ^d has range exponent at least d/(d+1)
AU - Ahmed Bou-Rabee
AU - Yuval Peres
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00090
UR - https://arxiv.org/abs/2610.00090
AB - In once-reinforced random walk, introduced by Davis (1990), every edge starts with weight one and is assigned weight β≥1 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 ℤ^d. We prove the corresponding lower bound in every dimension d≥2 and for every reinforcement parameter β.
ER -