The random walk penalised by its range in dimensions d3
Annales Henri Lebesgue, Volume 4 (2021) , pp. 1-79.

Metadata

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(-|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.


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