We consider Gibbs distributions on the set of permutations of Zd\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb Z}^d$$\end{document} associated to the Hamiltonian H(σ):=∑xV(σ(x)-x)\documentclass[12pt]{minimal}
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\begin{document}$$H(\sigma ):=\sum _{x} {V}(\sigma (x)-x)$$\end{document}, where σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document} is a permutation and V:Zd→R\documentclass[12pt]{minimal}
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\begin{document}$${V}:{\mathbb Z}^d\rightarrow {\mathbb R}$$\end{document} is a strictly convex potential. Call finite-cycle those permutations composed by finite cycles only. We give conditions on V\documentclass[12pt]{minimal}
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\begin{document}$${V}$$\end{document} ensuring that for large enough temperature α>0\documentclass[12pt]{minimal}
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\begin{document}$$\alpha >0$$\end{document} there exists a unique infinite volume ergodic Gibbs measure μα\documentclass[12pt]{minimal}
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\begin{document}$$\mu ^\alpha $$\end{document} concentrating mass on finite-cycle permutations; this measure is equal to the thermodynamic limit of the specifications with identity boundary conditions. We construct μα\documentclass[12pt]{minimal}
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\begin{document}$$\mu ^{\alpha }$$\end{document} 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 τv\documentclass[12pt]{minimal}
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\begin{document}$$\tau _v$$\end{document} as the shift permutation τv(x)=x+v\documentclass[12pt]{minimal}
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\begin{document}$$\tau _v(x)=x+v$$\end{document}. In the Gaussian case V=‖·‖2\documentclass[12pt]{minimal}
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\begin{document}$${V}=\Vert \cdot \Vert ^2$$\end{document}, we show that for each v∈Zd\documentclass[12pt]{minimal}
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\begin{document}$$v\in {\mathbb Z}^d$$\end{document}, μvα\documentclass[12pt]{minimal}
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\begin{document}$$\mu ^\alpha _v$$\end{document} given by μvα(f)=μα[f(τv·)]\documentclass[12pt]{minimal}
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\begin{document}$$\mu ^\alpha _v(f)=\mu ^\alpha [f(\tau _v\cdot )]$$\end{document} is an ergodic Gibbs measure equal to the thermodynamic limit of the specifications with τv\documentclass[12pt]{minimal}
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\begin{document}$$\tau _v$$\end{document} boundary conditions. For a general potential V\documentclass[12pt]{minimal}
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\begin{document}$${V}$$\end{document}, we prove the existence of Gibbs measures μvα\documentclass[12pt]{minimal}
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\begin{document}$$\mu ^\alpha _v$$\end{document} when α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} is bigger than some v\documentclass[12pt]{minimal}
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\begin{document}$$v$$\end{document}-dependent value.
