Condensation of a self-attracting random walk

I will introduce a Gibbs distribution 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. This model can be thought of as a random walk version of the Wulff crystal problem in percolation or the Ising model.

In joint work with Ariel Yadin 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)}.

This talk is part of the Probability series.