Condensation of a self-attracting random walk

Nathanaël Berestycki, Ariel Yadin

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


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.

Original languageEnglish
Pages (from-to)835-861
Number of pages27
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Issue number2
StatePublished - 1 May 2019


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

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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