A limit theorem for the covariances of spins in the Sherrington-Kirkpatrick model with an external field. (Un théorème limite pour les covariances des spins dans le modèle de Sherrington-Kirkpatrick avec champ externe.) (French) Zbl 1115.82037

The author studies the Sherrington-Kirkpatrick model at high enough temperature (\(\beta\ll1\)) and non-zero magnetic field. He considers the spin-spin covariance function \[ \gamma_{i,j}:=\langle \sigma_i \sigma_j\rangle- \langle \sigma_i\rangle \langle \sigma_j\rangle, \] where \(\langle .\rangle\) is the desorder-dependent Gibbs average. Of course \(\gamma_{i,j}\) is a random variable (which depends also on \(N\), the total number of spins in the system). It is well known that \(\gamma_{i,j}\) vanishes for \(N\to\infty\) in the high-temperature region. The main result (Theorem 1.1) is that, for \(N\to\infty\), \(\sqrt N \gamma_{i,j}\) converges in distribution to a non-Gaussian limit random variable, whose law is explicitly given.
The proof is based on the cavity method i.e., essentially, induction over \(N\) with a careful control on error terms. The paper is written in French.


82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
60G15 Gaussian processes
60F05 Central limit and other weak theorems
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