Ordered overlaps in disordered mean-field models |
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Authors: | Francis Comets Amir Dembo |
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Affiliation: | (1) Université Paris 7 – Denis Diderot, Mathématiques, case 7012, 2 place Jussieu, 75251 Paris Cedex 05, France. e-mail: comets@math.jussieu.fr, FR;(2) Department of Mathematics and Department of Statistics, Stanford University, Stanford, CA 94305, USA. e-mail: amir@math.Stanford.edu, US |
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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. Received: 6 April 1999 / Revised version: 29 May 2000 /?Published online: 24 July 2001 |
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Keywords: | Mathematics Subject Classification (2000): 60F10 82B44 82D30 |
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