Fluctuations of the free energy and overlaps in the high temperature p-spin SK and Hopfield models

We study the fluctuations of the free energy and overlaps of n replicas for the p-spin Sherrington-Kirkptarick and Hopfield models of spin glasses in the high temperature phase. For the first model we show that at all inverse temperatures β smaller than Talagrand’s bound βp the free energy on the scale N1−(p−2)/2 converges to a Gaussian law with zero mean and variance βp!/2; and that the law of the overlaps σ ·σ′ = Ni=1 σiσ′ i of n replicas on the scale √ N under the product of Gibbs measures is asymptotically the one of n(n − 1)/2 independent standard Gaussian random variables. For the second model we prove that for all β and the load of the memory t with β(1 + √ t) < 1 the law of the overlaps of n replicas on the scale √ N under the product of Gibbs measures is asymptotically the one of n(n− 1)/2 independent Gaussian random variables with zero mean and variance (1− tβ2(1− β)−2)−1. AMS Subject Classification: 82B44