Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Condensation of a self-attracting random walk

Nathanaël Berestycki and Ariel Yadin

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We introduce a Gibbs measure on nearest-neighbour paths of length $t$ in the Euclidean $d$-dimensional lattice, where each path is penalised by a factor proportional to the size of its boundary and an inverse temperature $\beta $. We prove that, for all $\beta >0$, the random walk condensates to a set of diameter $(t/\beta )^{1/3}$ in dimension $d=2$, up to a multiplicative constant. In all dimensions $d\ge 3$, we also prove that the volume is bounded above by $(t/\beta )^{d/(d+1)}$ and the diameter is bounded below by $(t/\beta )^{1/(d+1)}$. Similar results hold for a random walk conditioned to have local time greater than $\beta $ everywhere in its range when $\beta $ is larger than some explicit constant, which in dimension two is the logarithm of the connective constant.


Nous introduisons une mesure de Gibbs sur les chemins de longueur $t$ dans le réseau Euclidien de dimension $d$, telle qu’un chemin donné est penalisé par un facteur proportionnel à la taille de sa frontière et l’inverse d’une température $\beta >0$. Nous montrons qu’en dimension $d=2$, la marche aléatoire se condense dans un ensemble de diamètre $(t/\beta )^{1/3}$ à une constante multiplicative près. En dimensions $d\ge 3$, nous montrons que la marche occupe un volume inférieur à $(t/\beta )^{d/(d+1)}$ et son diamètre est au moins $(t/\beta )^{1/(d+1)}$. Des résultats similaires sont obtenus pour une marche aléatoire conditionnée à avoir un temps local supérieur à $\beta $ en chaque point visité, pourvu que $\beta $ soit supérieur à une constante explicite qui en deux dimensions est égale au logarithme de la constante de connectivité.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 2 (2019), 835-861.

Received: 19 October 2014
Revised: 6 November 2017
Accepted: 18 March 2018
First available in Project Euclid: 14 May 2019

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Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J27: Continuous-time Markov processes on discrete state spaces 60F10: Large deviations

Gibbs measure Condensation Self-attractive random walk Wulff crystal Large deviations Donsker–Varadhan principle


Berestycki, Nathanaël; Yadin, Ariel. Condensation of a self-attracting random walk. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 2, 835--861. doi:10.1214/18-AIHP900.

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