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Hypercontractivité pour des systèmes de spins de portée infinie
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  • Published: March 1995

Hypercontractivité pour des systèmes de spins de portée infinie

  • Etienne Laroche1 

Probability Theory and Related Fields volume 101, pages 89–132 (1995)Cite this article

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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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Authors and Affiliations

  1. Laboratoire de Statistique et Probabilités, Université Paul Sabatier, F-31062, Toulouse, France

    Etienne Laroche

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  1. Etienne Laroche
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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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  • Received: 22 October 1993

  • Revised: 01 July 1994

  • Issue Date: March 1995

  • DOI: https://doi.org/10.1007/BF01192197

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Mathematicals Subject Classification (1991)

  • 60K35
  • 82C20
  • 82C26
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