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Construction of pure states in mean field models for spin glasses
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  • Published: 17 November 2009

Construction of pure states in mean field models for spin glasses

  • Michel Talagrand1 

Probability Theory and Related Fields volume 148, pages 601–643 (2010)Cite this article

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Abstract

If a mean field model for spin glasses is generic in the sense that it satisfies the extended Ghirlanda–Guerra identities, and if the law of the overlaps has a point mass at the largest point q* of its support, we prove that one can decompose the configuration space into a sequence of sets (A k ) such that, generically, the overlap of two configurations is equal to q* if and only if they belong to the same set A k . For the study of the overlaps each set A k can be replaced by a single point. Combining this with a recent result of Panchenko (A connection between Ghirlanda–Guerra identities and ultrametricity. Ann Probab (2008, to appear)) this proves that if the overlaps take only finitely many values, ultrametricity occurs. We give an elementary, self-contained proof of this result based on simple inequalities and an averaging argument.

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Authors and Affiliations

  1. Equipe d’Analyse, Institut de Mathématiques, UMR 7586, CNRS, Paris, France

    Michel Talagrand

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  1. Michel Talagrand
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Correspondence to Michel Talagrand.

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Talagrand, M. Construction of pure states in mean field models for spin glasses. Probab. Theory Relat. Fields 148, 601–643 (2010). https://doi.org/10.1007/s00440-009-0242-6

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  • Received: 13 November 2008

  • Revised: 12 July 2009

  • Published: 17 November 2009

  • Issue Date: November 2010

  • DOI: https://doi.org/10.1007/s00440-009-0242-6

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Mathematics Subject Classification (2000)

  • Primary 80 D 30
  • Secondary 60 G 15
  • 28 C 20
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