Condensation of a self-attracting random walk

Research output: Contribution to journalArticlepeer-review

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.

Original language English 835-861 27 Annales de l'institut Henri Poincare (B) Probability and Statistics 55 2 https://doi.org/10.1214/18-AIHP900 Published - 1 May 2019

• Condensation