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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.

MSC:

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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References:

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