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

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.