@article{57235baa1e3f48cda0e5bde6031eea13,

title = "Condensation of a self-attracting random walk",

abstract = " 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 β. We prove that, for all β > 0, the random walk condensates to a set of diameter (t/β) 1 / 3 in dimension d = 2, up to a multiplicative constant. In all dimensions d ≥ 3, we also prove that the volume is bounded above by (t/β) d/(d +1 ) and the diameter is bounded below by (t/β) 1 /(d+1) Similar results hold for a random walk conditioned to have local time greater than β everywhere in its range when β is larger than . some explicit constant, which in dimension two is the logarithm of the connective constant. ",

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

author = "Nathana{\"e}l Berestycki and Ariel Yadin",

note = "Funding Information: 1Supported in part by EPSRC grants EP/GO55068/1 and EP/I03372X/1. This work started when AY was a Herschel Smith postdoctoral fellow in 2009–2010 at the Statistical Laboratory, University of Cambridge. We gratefully acknowledge the financial support of the Herschel Smith fund and EPSRC grant EP/GO55068/1 as well as EP/L018896/1 and EP/I03372X/1. The first author is grateful for the hospitality of the Theory Group at Microsoft Research, where part of this work was carried out. He would also like to thank Omer Angel, Ori Gurel-Gurevitch, Yuval Peres and Ofer Zeitouni for useful conversations, and Tom Begley for the pictures. We are very grateful to two anonymous referees for their comments which improved the presentation and pointed out some mistakes in earlier versions of the paper. Funding Information: This work started when AY was a Herschel Smith postdoctoral fellow in 2009–2010 at the Statistical Laboratory, University of Cambridge. We gratefully acknowledge the financial support of the Herschel Smith fund and EPSRC grant EP/GO55068/1 as well as EP/L018896/1 and EP/I03372X/1. The first author is grateful for the hospitality of the Theory Group at Microsoft Research, where part of this work was carried out. He would also like to thank Omer Angel, Ori Gurel-Gurevitch, Yuval Peres and Ofer Zeitouni for useful conversations, and Tom Begley for the pictures. We are very grateful to two anonymous referees for their comments which improved the presentation and pointed out some mistakes in earlier versions of the paper. Funding Information: 1Supported in part by EPSRC grants EP/GO55068/1 and EP/I03372X/1. Publisher Copyright: {\textcopyright} Association des Publications de l{\textquoteright}Institut Henri Poincar{\'e}, 2019",

year = "2019",

month = may,

day = "1",

doi = "10.1214/18-AIHP900",

language = "English",

volume = "55",

pages = "835--861",

journal = "Annales de l'institut Henri Poincare (B) Probability and Statistics",

issn = "0246-0203",

publisher = "Institute of Mathematical Statistics",

number = "2",

}