Finite Cycle Gibbs Measures on Permutations of ${{\mathbb Z}^d}$

Abstract

We consider Gibbs distributions on the set of permutations of ${\mathbb Z}^d$ associated to the Hamiltonian $H(\sigma ):=\sum _{x} {V}(\sigma (x)-x)$, where $\sigma$ is a permutation and ${V}:{\mathbb Z}^d\rightarrow {\mathbb R}$ 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 $\alpha >0$ there exists a unique infinite volume ergodic Gibbs measure $\mu ^\alpha$ concentrating mass on finite-cycle permutations; this measure is equal to the thermodynamic limit of the specifications with identity boundary conditions. We construct $\mu ^{\alpha }$ 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 continuous-time birth and death process of cycles interacting by exclusion, an approach proposed by Fernández, Ferrari and Garcia. Define $\tau _v$ as the shift permutation $\tau _v(x)=x+v$. In the Gaussian case ${V}=\Vert \cdot \Vert ^2$, we show that for each $v\in {\mathbb Z}^d$, $\mu ^\alpha _v$ given by $\mu ^\alpha _v(f)=\mu ^\alpha [f(\tau _v\cdot )]$ is an ergodic Gibbs measure equal to the thermodynamic limit of the specifications with $\tau _v$ boundary conditions. For a general potential ${V}$, we prove the existence of Gibbs measures $\mu ^\alpha _v$ when $\alpha$ is bigger than some $v$-dependent value.