Fluctuations of the Local Magnetic Field in Frustrated Mean-field Ising Models

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. 1. Frustrated Ising models and the local field distribution The celebrated Sherrington-Kirkpatrick model of a spin glass is given by the Hamiltonian HSK = 1 √ N N ∑ i,j=1 Ji,jsisj , (1.1) where the si = ±1 are Ising spins and the interaction parameters Ji,j are i.i.d. random variables with Gaussian distribution. It was proposed and solved in [1, 2] by Sherrington and Kirkpatrick using the replica trick. However, their solution is flawed because it predicts negative entropy at low temperatures. An alternative solution scheme was proposed by Parisi [3, 4], which is generally regarded as being correct. However, it also involves the mathematically dubious replica trick and the mathematical status of the solution is therefore still unclear. Indeed, this model presents a considerable challenge to mathematicians [9]. Nevertheless, some progress has been made. Aizenman, Lebowitz and Ruelle [6] proved that, in the absence of an external field, the Sherrington-Kirkpatrick (SK) solution is correct in the high temperature domain. Pastur and Schcherbina [7] proved that the SK solution is correct unless the Edwards-Anderson order parameter is not self-averaging (which implies the latter). Guerra [8] derived a beautiful inequality which implies that the the SK solution is correct in the high-temperature domain, 2000 Mathematics Subject Classification. Primary: 82B44, Secondary 60B10, 60B05.