Quadratic replica coupling in the Sherrington–Kirkpatrick mean field spin glass model

Abstract

We develop a very simple method to study the high temperature, or equivalently high external field, behavior of the Sherrington–Kirkpatrick mean field spin glass model. The basic idea is 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 reproduce the well known validity of the annealed approximation, up to the known critical value for the temperature. In the case of nontrivial external field, we can prove the validity of the Sherrington–Kirkpatrick replica symmetric solution up to a line, which falls short of the Almeida–Thouless line, associated to the onset of the spontaneous replica symmetry breaking, in the Parisi ansatz. The main difference with the method, recently developed by Michel Talagrand, is that we employ a quadratic coupling, and not a linear one. The resulting flow equations, with respect to the parameters of the model, turn out to be very simple, and the parameter region, where the method works, can be easily found in explicit terms. As a straightforward application of cavity methods, we show also how to determine free energy and overlap fluctuations, in the region where replica symmetry has been shown to hold. It is a major open problem to give a rigorous mathematical treatment of the transition to replica symmetry breaking, necessarily present in the model.