We consider Gibbs distributions on the set of permutations of $${\mathbb Z}^d$$ Z d associated to the Hamiltonian $$H(\sigma ):=\sum _{x} {V}(\sigma (x)-x)$$ H ( σ ) : = ∑ x V ( σ ( x ) - x ) , where $$\sigma $$ σ is a permutation and $${V}:{\mathbb Z}^d\rightarrow {\mathbb R}$$ V : Z d → R is a strictly convex potential. Call finite-cycle those permutations composed by finite cycles only. We give conditions on $${V}$$ V ensuring that for large enough temperature $$\alpha >0$$ α > 0 there exists a unique infinite volume ergodic Gibbs measure $$\mu ^\alpha $$ μ α concentrating mass on finite-cycle permutations; this measure is equal to the thermodynamic limit of the specifications with identity boundary conditions. We construct $$\mu ^{\alpha }$$ μ α 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 $$\tau _v$$ τ v as the shift permutation $$\tau _v(x)=x+v$$ τ v ( x ) = x + v . In the Gaussian case $${V}=\Vert \cdot \Vert ^2$$ V = ‖ · ‖ 2 , we show that for each $$v\in {\mathbb Z}^d$$ v ∈ Z d , $$\mu ^\alpha _v$$ μ v α given by $$\mu ^\alpha _v(f)=\mu ^\alpha [f(\tau _v\cdot )]$$ μ v α ( f ) = μ α [ f ( τ v · ) ] is an ergodic Gibbs measure equal to the thermodynamic limit of the specifications with $$\tau _v$$ τ v boundary conditions. For a general potential $${V}$$ V , we prove the existence of Gibbs measures $$\mu ^\alpha _v$$ μ v α when $$\alpha $$ α is bigger than some $$v$$ v -dependent value.