The random walk penalised by its range in dimensions $d\geq 3$
Résumé
We study a self-attractive random walk such that each trajectory of length is penalised by a factor proportional to , where 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 , for some explicit constant . This proves a conjecture of Bolthausen [Bol94] who obtained this result in the case .
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