The replica symmetric region in the Sherrington-Kirkpatrick mean field   spin glass model. The Almeida-Thouless line
  In previous work, we have developed a simple method to study the behavior of the Sherrington-Kirkpatrick mean field spin glass model for high temperatures, or equivalently for high external fields. The basic idea was to couple two different replicas with a quadratic term, trying to push out the two replica overlap from its replica symmetric value. In the case of zero external field, our results reproduced the well known validity of the annealed approximation, up to the known critical value for the temperature. In the case of nontrivial external field, our method could prove the validity of the Sherrington-Kirkpatrick replica symmetric solution up to a line, which fell short of the Almeida-Thouless line, associated to the onset of the spontaneous replica symmetry breaking, in the Parisi Ansatz. Here, we make a strategic improvement of the method, by modifying the flow equations, with respect to the parameters of the model. We exploit also previous results on the overlap fluctuations in the replica symmetric region. As a result, we give a simple proof that replica symmetry holds up to the critical Almeida-Thouless line, as expected on physical grounds. Our results are compared with the characterization of the replica symmetry breaking line previously given by Talagrand. We outline also a possible extension of our methods to the broken replica symmetry region. 
Francesco Guerra