A field-theoretical approach to the spin glass transition: models with long but finite interaction range

We study spin glasses with the Kac-type interaction potential for small but finite inverse interaction range γ. Using the theoretical set-up of coupled replicas, through the replica method we argue that the probability of overlap profiles can be expressed for small γ through a large deviation functional. This result is supported by rigorous arguments, showing that the large deviation functional provides at least upper bounds for the probability. Finally we analyse the rate function, in the vicinity of the critical point Tc = 1, h = 0 of mean field theory, and we study the free energy cost of overlap interfaces, assuming the validity of a gradient expansion for the rate functional.