The Ising-Sherrington-Kirpatrick model in a magnetic field at high temperature (Q2481424)

The authors consider a \(d\) -dimensional Ising model in a magnetic field \(h\),perturbed by a mean field interaction of spin-glass type. The Hamiltonian contains two parameters, \(\beta\) and \(\kappa\), which play the role of two inverse temperatures. When \(\beta=0\) the model reduces to the Ising model at temperature \(\frac{1}{\kappa}\), while for \(\kappa=0\) one recovers the Sherrington-Kirkpatrick model at temperature \(\frac{1}{\beta}.\) It is proved existence of the pressure in the limit of an infinite box. When both Ising and Sherrington-Kirkpatrick temperatures are high enough, it is proved that the value of the pressure is given by a suitable replica symmetric solution and the fluctuations of the pressure are of order of the inverse of the square of the volume with a normal distribution in the limit. In this regime, the pressure can be expressed in terms of random field Ising models.