Résumé
Nous considérons un système de spins surℤ d. Nous prouvons l'équivalence entre premièrement une condition faible de mélange deuxièmement le contrôle du trou dans le spectre et troisièmement celui de la constante de Sobolev logarithmique pour des potentiels de Gibbs de portée non nécessairement finie. Nous en tirons des conséquences sur la vitesse de convergence des semi-groupes dansL 2 et sur la décroissance des corrélations: il n'y a pas de régime intermédiaire entre un taux algébrique ent −2d (resp. |j-k|−2d) et un taux exponentiel. Les résultats généraux sont montrés pour des spins à valeur dans une variété riemannienne compacte ou dans un espace fini.
Summary
We consider a spin system onℤ d. We prove the equivalence between first a weak mixing condition, secondly the controle of spectral gap and thirdly the controle of logarithmic Sobolev constants for non necessarily finite range Gibbs potentials. Hence we draw consequences concerning theL 2 decay to equilibrium and the correlations decay: there is no transitory rate between an algebraic decay ast −2d (resp. |j-k|−2d)and exponential decay. The general results are obtained for both continuous and discrete compact spins.
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Laroche, E. Hypercontractivité pour des systèmes de spins de portée infinie. Probab. Th. Rel. Fields 101, 89–132 (1995). https://doi.org/10.1007/BF01192197
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DOI: https://doi.org/10.1007/BF01192197
Mathematicals Subject Classification (1991)
- 60K35
- 82C20
- 82C26