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

Berestycki, Nathanaël; Cerf, Raphaël

doi:10.5802/ahl.66

The random walk penalised by its range in dimensions

d \geq 3

[La marche aléatoire pénalisée par son image en dimension

d \geq 3

]

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)

Annales Henri Lebesgue, Tome 4 (2021), pp. 1-79.

Résumé
Abstract

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 $d = 2$ .

Reçu le : 2019-01-11
Révisé le : 2019-12-10
Accepté le : 2020-02-25
Publié le : 2021-01-18

DOI : 10.5802/ahl.66

Classification : 60F05, 60G50, 82B41
Mots clés : Random walk, Faber–Krahn, large deviations

Affiliations des auteurs :

Berestycki, Nathanaël ¹ ; Cerf, Raphaël ²

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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://www.numdam.org/articles/10.5802/ahl.66/

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