  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.
