Fluctuations of the Free Energy and Overlaps in the High-Temperature p-Spin SK and Hopfield Models

#### I.A. Kurkova

2005, v.11, Issue 1, 55-80

ABSTRACT

We study the fluctuations of the free energy and overlaps of $n$ replicas for the p-spin Sherrington - Kirkpatrick and Hopfield models of spin glasses in the high temperature phase. For the first model we show that at all inverse temperatures $\beta$ smaller than Talagrand's bound $\beta_p$ the free energy on the scale $N^{1-(p-2)/2}$ converges to a Gaussian law with zero mean and variance $\b^4 p!/2$; and that the law of the overlaps $\s\cdot \s'=\sum_{i=1}^{N}\s_i\s'_i$ of $n$ replicas on the scale $\sqrt{N}$ 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 $\beta$ and the load of the memory $t$ with $\beta(1+\sqrt{t})<1$ the law of the overlaps of $n$ replicas on the scale $\sqrt{N}$ under the product of Gibbs measures is asymptotically the one of $n(n-1)/2$ independent Gaussian random variables with zero mean and variance $(1-t\b^2(1-\b)^{-2})^{-1}$.

Keywords: spin glasses,Sherrington - Kirkpatrick model,p-spin model,Hopfield model,overlap,free energy,martingales

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