The random walk penalised by its range in dimensions d3
[La marche aléatoire pénalisée par son image en dimension d3]
Annales Henri Lebesgue, Tome 4 (2021), pp. 1-79.

Nous étudions une marche aléatoire auto-attractive, chaque trajectoire de longueur N est pénalisée par un facteur proportionnel à exp(-|R N |), où R N est l’ensemble des sites visités par la marche. Nous montrons que l’image d’une telle marche aléatoire est proche d’une boule Euclidienne dont le rayon est approximativement ρ d N 1/(d+2) , avec une valeur explicite de la constante ρ d >0. Nous prouvons ainsi une conjecture de Bolthausen [Bol94], qui a obtenu ce résultat dans le cas d=2.

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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DOI : 10.5802/ahl.66
Classification : 60F05, 60G50, 82B41
Mots clés : Random walk, Faber–Krahn, large deviations
Berestycki, Nathanaël 1 ; Cerf, Raphaël 2

1 Universität Wien (Austria) On leave from the University of Cambridge, (UK)
2 Ecole Normale Supérieure, PSL University, CNRS, DMA,75005, Paris, (France) Université Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay, 91405, Orsay, (France)
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Berestycki, Nathanaël; Cerf, Raphaël. The random walk penalised by its range in dimensions $d\ge 3$. Annales Henri Lebesgue, Tome 4 (2021), pp. 1-79. doi : 10.5802/ahl.66. http://archive.numdam.org/articles/10.5802/ahl.66/

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