Finite cycle Gibbs measures on permutations of $\mathbb Z^d$

TL;DR

This paper studies Gibbs measures on permutations of bZ^d with convex potentials, establishing existence, uniqueness, and ergodic properties of finite-cycle permutation measures at high temperature, and analyzing their translation-invariant variants.

Contribution

It introduces a new Markov process approach to construct and analyze Gibbs measures on permutations with convex potentials, including translation-invariant cases.

Findings

Existence of unique infinite-volume Gibbs measures for large lpha.

Construction of Gibbs measures as invariant measures of a cycle-based Markov process.

Identification of translation-invariant Gibbs measures in the Gaussian case.

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.