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 $V:\mathbb Z^d\to\mathbb R$ 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 $\alpha>0$ 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 Fern\'andez, Ferrari and Garcia. Define $\tau_v$ as the shift permutation $\tau_v(x)=x+v$. In the Gaussian case $V=|\cdot|^2$, 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.