Condensation of a self-attracting random walk

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.