The random walk penalised by its range in dimensions $d\ge 3$
Annales Henri Lebesgue, Volume 4 (2021) , pp. 1-79.

KeywordsRandom walk, Faber–Krahn, large deviations

### Abstract

We study a self-attractive random walk such that each trajectory of length $N$ is penalised by a factor proportional to $exp\left(-|{R}_{N}|\right)$, 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 ${\rho }_{d}{N}^{1/\left(d+2\right)}$, for some explicit constant ${\rho }_{d}>0$. This proves a conjecture of Bolthausen [] who obtained this result in the case $d=2$.

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