A limit theorem for spin covariances in the Sherrington-Kirkpatrick model with an external field (Q2583828)

This note focuses on the asymptotic fluctuations of a two-site correlation function \[ \gamma_{i,j}=\langle\sigma_i\sigma_j\rangle -\langle\sigma_i\rangle\langle\sigma_j\rangle, \] where \(i,j\) are distinct indices in \(\{ 1,2,\ldots ,N\} \) whereas \(\langle\cdot\rangle\) denotes an expected value taken with respect to the equilibrium measure of the Sherrington-Kirkpatrick model, a random probability measure on the simplex of all spin configurations \(\sigma\in\{\pm 1\}^N\). More precisely, the S-K model is considered in the high temperature regime and in the presence of an external magnetic field; \(\beta\) and \(h\) denoting the inverse temperature and external field parameters, there exists a nonnegative real number \(q_2\) for which \( E[ \tanh^2(Y)]=q_2\) when \(Y=\beta\sqrt{q_2}z+h\) and \(z\) is a standard Gaussian random variable. Further quantities are introduced, namely the random variable \(U=1-\tanh^2(Y)\) as well as \(C=\beta/\sqrt{1-\beta^2E[U^2]}\), and the main result may then be stated as follows: for small values of the parameter \(\beta\) (\(\beta\leq\beta_0\)) and for every fixed integer \(p\geq 1\), the moment of order \(p\) associated with the random variable \(\sqrt{N}\gamma_{i,j}\) converges to the \(p\)th moment of \(V=C z U_1U_2\), where \(z,U_1\) and \(U_2\) are independent, \(U_1\) and \(U_2\) denoting two copies of the random variable \(U\). The high temperature fluctuations of \(\gamma_{i,j}\) on an \(N^{-1/2}\) scale thus have a non-Gaussian limiting distribution, as soon as \(h\neq 0\). An outline of the proof is given, where the cavity method is being used together with earlier results on the high temperature fluctuations of the overlaps [cf. \textit{M. Talagrand}, ``Spin glasses: A challenge to mathematicians. Cavity and mean field models'' (2003; Zbl 1033.82002)].