Bose-Einstein condensation and infinite cycles in random permutation

I present a model of random permutations on a set with spatial structure. The probability of obtaining a given permutation is determined by a Gibbs factor, and the energy is higher when the permutation contains more jumps between distant points of the underlying set. So, the jump length of a typical random permutation will be small. For this model I show the existence of a phase transition: Depending on the density of the points forming the spatial structure, there either exist exclusively finite cycles (for low density), or a coexistence of finite and macroscopic cycles (for high density). The physical relevance of the model comes from its connections to Bose-Einstein condensation; I will briefly explain these connections and highlight open question. This is joint work with Daniel Ueltschi.