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A general form of certain mean field models for spin glasses
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  • Published: 08 December 2007

A general form of certain mean field models for spin glasses

  • Michel Talagrand1 

Probability Theory and Related Fields volume 143, pages 97–111 (2009)Cite this article

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  • 4 Citations

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Abstract

Given numbers a ij ≥ 0 for 1 ≤ i  <  j ≤ N, and given numbers b i ≥ 0, i ≤ N, we consider the random Hamiltonian \(\sum_{i,j \le N} \sqrt{a_{ij}} g_{ij} \sigma_i \sigma_j + \sum_{i \le N} \sqrt{b_i} g_i \sigma_i\) , where g i , g ij denote independent standard normal r.v., and where σ i = ± 1. We give sufficient conditions on the coefficients a ij for the system governed by this Hamiltonian to exhibit “high-temperature behavior”. There results extend known facts concerning the behavior of the Sherrington-Kirkpatrick model at “very high-temperature”. In a similar manner we give a general form of the “perceptron model”.

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References

  1. Talagrand M. (2003). Spin Glasses, a Challenge for Mathematicians. Springer, Heidelberg

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Authors and Affiliations

  1. Institut de Mathématiques, Université Paris VI, 4 Place Jussieu, 75230, Paris Cedex 05, France

    Michel Talagrand

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  1. Michel Talagrand
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Correspondence to Michel Talagrand.

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Talagrand, M. A general form of certain mean field models for spin glasses. Probab. Theory Relat. Fields 143, 97–111 (2009). https://doi.org/10.1007/s00440-007-0121-y

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  • Received: 24 January 2007

  • Revised: 02 November 2007

  • Published: 08 December 2007

  • Issue Date: January 2009

  • DOI: https://doi.org/10.1007/s00440-007-0121-y

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Mathematics Subject Classifications (2000)

  • Primary: 82B44
  • Secondary: 60G15
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