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Renormalization Group in the Uniqueness Region: Weak Gibbsianity and Convergence

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Abstract

We analyze the block averaging transformation applied to lattice gas models with short range interaction in the uniqueness region below the critical temperature. We prove weak Gibbsianity of the renormalized measure and convergence of the renormalized potential in a weak sense. Since we are arbitrarily close to the coexistence region we have a diverging characteristic length of the system: the correlation length or the critical length for metastability, or both. Thus, to perturbatively treat the problem we have to use a scale–adapted expansion. Moreover, such a model below the critical temperature resembles a disordered system in the presence of Griffiths' singularity. Then the cluster expansion that we use must be graded with its minimal scale length diverging when the coexistence line is approached.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma La Sapienza, Piazzale Aldo Moro 2, 00185, Roma, Italy

    Lorenzo Bertini

  2. Dipartimento Me. Mo. Mat., Università di Roma La Sapienza, Via A. Scarpa 16, 00161, Roma, Italy

    Emilio N.M. Cirillo

  3. Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 00133, Roma, Italy

    Enzo Olivieri

Authors
  1. Lorenzo Bertini
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  2. Emilio N.M. Cirillo
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  3. Enzo Olivieri
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Correspondence to Lorenzo Bertini.

Additional information

Communicated by J.Z. Imbrie

The authors acknowledge the support of Cofinanziamento MIUR.

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Bertini, L., Cirillo, E. & Olivieri, E. Renormalization Group in the Uniqueness Region: Weak Gibbsianity and Convergence. Commun. Math. Phys. 261, 323–378 (2006). https://doi.org/10.1007/s00220-005-1399-1

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