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Renormalization-Group Transformations Under Strong Mixing Conditions: Gibbsianness and Convergence of Renormalized Interactions

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Abstract

In this paper we study a renormalization-group map: the block averaging transformation applied to Gibbs measures relative to a class of finite-range lattice gases, when suitable strong mixing conditions are satisfied. Using a block decimation procedure, cluster expansion, and detailed comparison between statistical ensembles, we are able to prove Gibbsianness and convergence to a trivial (i.e., Gaussian and product) fixed point. Our results apply to the 2D standard Ising model at any temperature above the critical one and arbitrary magnetic field.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma “La Sapienza,”, 00185, Rome, Italy

    Lorenzo Bertini

  2. CMI, Université de Provence, 13453, Marseille, France

    Emilio N. M. Cirillo

  3. Dipartimento di Matematica, II Università di Roma Tor Vergata, 0133, Rome, Italy

    Enzo Olivieri

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  1. Lorenzo Bertini
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Bertini, L., Cirillo, E.N.M. & Olivieri, E. Renormalization-Group Transformations Under Strong Mixing Conditions: Gibbsianness and Convergence of Renormalized Interactions. Journal of Statistical Physics 97, 831–915 (1999). https://doi.org/10.1023/A:1004620929047

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