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Article Dans Une Revue Annales Henri Lebesgue Année : 2021

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 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 [Bol94] who obtained this result in the case d=2.
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Dates et versions

hal-03998535 , version 1 (15-02-2024)

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Paternité

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Nathanael Berestycki, Raphaël Cerf. The random walk penalised by its range in dimensions $d\geq 3$. Annales Henri Lebesgue, 2021, 4, pp.1-79. ⟨10.5802/ahl.66⟩. ⟨hal-03998535⟩
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