The random walk penalised by its range in dimensions $d\geq 3$ Article Swipe

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Nathanaël Berestycki , Raphaël Cerf ·

YOU? · · 2018 · Open Access · · DOI: https://doi.org/10.48550/arxiv.1811.04700 · OA: W2900257979

We study a self-attractive random walk such that each trajectory of length $N$ is penalised by a factor proportional to $\exp ( - |R_N|)$, where $R_N$ is the set of sites visited by the walk. We show that the range of such a walk is close to a solid Euclidean ball of radius approximately $ρ_d N^{1/(d+2)}$, for some explicit constant $ρ_d >0$. This proves a conjecture of Bolthausen who obtained this result in the case $d=2$.