Ordered overlaps in disordered mean-field models

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