The random walk penalised by its range in dimensions $d\ge 3$

Abstract

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 ${\rho }_d{N}^{1/(d+2)}$, for some explicit constant ${\rho }_d>0$. This proves a conjecture of Bolthausen who obtained this result in the case $d=2$.

Revised version, local errors and typos corrected

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