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Graded Cluster Expansion for Lattice Systems

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Abstract

In this paper we develop a general theory which provides a unified treatment of two apparently different problems. The weak Gibbs property of measures arising from the application of Renormalization Group maps and the mixing properties of disordered lattice systems in the Griffiths’ phase. We suppose that the system satisfies a mixing condition in a subset of the lattice whose complement is sparse enough namely, large regions are widely separated. We then show how it is possible to construct a convergent multi-scale cluster expansion.

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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. Graded Cluster Expansion for Lattice Systems. Commun. Math. Phys. 258, 405–443 (2005). https://doi.org/10.1007/s00220-005-1360-3

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