## 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

### Keywords:

Sherrington-Kirkpatrick model; central limit theorem
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### References:

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