Finite Cycle Gibbs Measures on Permutations of Zd

Abstract

We consider Gibbs distributions on the set of permutations of Zd associated to the Hamiltonian H(σ ) := x V(σ (x) − x), where σ is a permutation and V : Zd → R is a strictly convex potential. Call finite-cycle those permutations composed by finite cycles only. We give conditions on V ensuring that for large enough temperature α > 0 there exists a unique infinite volume ergodic Gibbs measure μα concentrating mass on finite-cycle permutations; this measure is equal to the thermodynamic limit of the specifications with identity boundary conditions. We construct μα as the unique invariant measure of a Markov process on the set of finite-cycle permutations that can be seen as a loss-network, a continuoustime birth and death process of cycles interacting by exclusion, an approach proposed by Fernández, Ferrari and Garcia. Define τv as the shift permutation τv(x) = x + v. In the Gaussian case V =·2, we show that for each v ∈ Zd , μα v given by μα v ( f ) = μα[ f (τv·)] is an ergodic Gibbs measure equal to the thermodynamic limit of the specifications with τv boundary conditions. For a general potential V, we prove the existence of Gibbs measures μα v when α is bigger than some v-dependent value.