Positive-overlap transition and critical exponents in mean field spin glasses

In this paper we obtain two results for the Sherrington-Kirkpatrick (SK) model, and we show that they both emerge from a single approach. First, we prove that the average of the overlap takes positive values when it is nonzero. More specifically, the average of the overlap, which is naively expected to take values in the whole interval [-1,+1], becomes positive if we 'first' apply an external field, so as to destroy the gauge invariance of the model, and 'then' remove it in the thermodynamic limit. This phenomenon emerges at the critical point. This first result is weaker than the one obtained by Talagrand (not limited to the average of the overlap), but we show here that, at least on average, the overlap is proven to be non-negative with no use of the Ghirlanda-Guerra identities. The latter are instead needed to obtain the second result, which is to control the behaviour of the overlap at the critical point: we find the critical exponents of all the overlap correlation functions.