Limit shape and range upper bound for once-reinforced random walk

Establish the matching upper bound for the expected range and determine the limit shape of once-reinforced random walk on \(\mathbb{Z}^d\) for \(d\geq2\), thereby resolving the conjectural behavior beyond the known lower bound on the expected range.

Background

The paper compares the centrally excited walk with once-reinforced random walk, for which the radius $n^{1/(d+1)}$ and range size $n^{d/(d+1)}$ are predicted. For once-reinforced random walk on $\mathbb{Z}^d$, $d\geq2$, prior work establishes only a lower bound of order $n^{d/(d+1)}$ on the expected range for every reinforcement parameter. The matching upper bound and a deterministic limit-shape theorem remain unresolved.