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Ordered overlaps in disordered mean-field models
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  • Published: September 2001

Ordered overlaps in disordered mean-field models

  • Francis Comets1 &
  • Amir Dembo2 

Probability Theory and Related Fields volume 121, pages 1–29 (2001)Cite this article

Abstract.

Let M(N) be a sequence of integers with M→∞ as N→∞ and M=o(N). For bounded i.i.d. r.v. ξ i k and bounded i.i.d. r.v. σ i , we study the large deviation of the family of (ordered) scalar products X k=N −1∑ i =1 N σ i ξ i k,k≤M, under the distribution conditioned on the ξ i k's. To get a full large deviation principle, it is necessary to specify also the total norm(∑ k ≤ M (X k)2)1/2, which turns to be associated with some extra Gaussian distribution. Our results apply to disordered, mean-field systems, including generalized Hopfield models in the regime of a sublinear number of patterns. We build also a class of examples where this norm is the crucial order parameter.

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Authors and Affiliations

  1. Université Paris 7 – Denis Diderot, Mathématiques, case 7012, 2 place Jussieu, 75251 Paris Cedex 05, France. e-mail: comets@math.jussieu.fr, , , , , , FR

    Francis Comets

  2. Department of Mathematics and Department of Statistics, Stanford University, Stanford, CA 94305, USA. e-mail: amir@math.Stanford.edu, , , , , , US

    Amir Dembo

Authors
  1. Francis Comets
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  2. Amir Dembo
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Received: 6 April 1999 / Revised version: 29 May 2000 /¶Published online: 24 July 2001

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Comets, F., Dembo, A. Ordered overlaps in disordered mean-field models. Probab Theory Relat Fields 121, 1–29 (2001). https://doi.org/10.1007/PL00008794

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  • Issue Date: September 2001

  • DOI: https://doi.org/10.1007/PL00008794

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  • Mathematics Subject Classification (2000): 60F10, 82B44, 82D30
  • Key words or phrases: Large deviations – Disordered systems – Spin glass
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