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Fluctuations of the free energy and overlaps in the high-temperature \(p\)-spin SK and Hopfield models. (English) Zbl 1121.82023

Summary: 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 \(\beta^4 p!/2\); and that the law of the overlaps \(\sigma\cdot \sigma'=\sum_{i=1}^{N}\sigma_i\sigma'_i\) 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 \((1-t\beta^2(1-\beta)^{-2})^{-1}\).

MSC:

82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
60F99 Limit theorems in probability theory
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