Kurkova, I. Fluctuations of the free energy and overlaps in the high-temperature \(p\)-spin SK and Hopfield models. (English) Zbl 1121.82023 Markov Process. Relat. Fields 11, No. 1, 55-80 (2005). 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 \((1-t\beta^2(1-\beta)^{-2})^{-1}\). Cited in 1 Document MSC: 82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics 60F99 Limit theorems in probability theory Keywords:spin glasses; Sherrington-Kirkpatrick model; \(p\)-spin model; Hopfield model; overlap; free energy; martingales PDFBibTeX XMLCite \textit{I. Kurkova}, Markov Process. Relat. Fields 11, No. 1, 55--80 (2005; Zbl 1121.82023)