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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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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DOI: https://doi.org/10.1007/PL00008794
- Mathematics Subject Classification (2000): 60F10, 82B44, 82D30
- Key words or phrases: Large deviations – Disordered systems – Spin glass