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 $H(σ):=\sum_{x} V(σ(x)-x)$, where $σ$ is a permutation and $V:\mathbb Z^d\to\mathbb R$ 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 $α>0$ there exists a unique infinite volume ergodic Gibbs measure $μ^α$ concentrating mass on finite-cycle permutations; this measure is equal to the thermodynamic limit of the specifications with identity boundary conditions. We construct $μ^α$ 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 $τ_v$ as the shift permutation $τ_v(x)=x+v$. In the Gaussian case $V=\|\cdot\|^2$, we show that for each $v\in\mathbb Z^d$, $μ^α_v$ given by $μ^α_v(f)=μ^α[f(τ_v\cdot)]$ is an ergodic Gibbs measure equal to the thermodynamic limit of the specifications with $τ_v$ boundary conditions. For a general potential $V$, we prove the existence of Gibbs measures $μ^α_v$ when $α$ is bigger than some $v$-dependent value.