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Fluctuations of the local magnetic field in frustrated mean-field Ising models

Dukes, W.M.B. and Dorlas, Tony (2004) Fluctuations of the local magnetic field in frustrated mean-field Ising models. Markov Processes and Related Fields, 10 (4). pp. 585-606. ISSN 1024-2953

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Abstract

We consider fluctuations of the local magnetic field in frustrated mean-field Ising models. Frustration can come about due to randomness of the interaction as in the Sherrington - Kirkpatrick model, or through fixed interaction parameters but with varying signs. We consider central limit theorems for the fluctuation of the local magnetic field values w.r.t. the a priori spin distribution for both types of models. We show that, in the case of the Sherrington - Kirkpatrick model there is a central limit theorem for the local magnetic field, a.s. with respect to the randomness. On the other hand, we show that, in the case of non-random frustrated models, there is no central limit theorem for the distribution of the values of the local field, but that the probability distribution of this distribution does converge. We compute the moments of this probability distribution on the space of measures and show in particular that it is not Gaussian.