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 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 $v\in {\mathbb{Z}}^d$, ${\mu }_v^{\alpha }$ given by ${\mu }_v^{\alpha }(f)={\mu }^{\alpha }[f({\tau }_v\cdot )]$ 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 ${\mu }_v^{\alpha }$ when $\alpha$ is bigger than some $v$-dependent value.

References in corpus (1)

• Spatial random permutations and infinite cycles