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

Abstract

In this paper we obtain two results for the Sherrington–Kirkpatrick (SK) model, and weshow that they both emerge from a single approach. First, we prove that the average of theoverlap takes positive values when it is non-zero. More specifically, the averageof 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 gaugeinvariance of the model, and ‘then’ remove it in the thermodynamic limit. Thisphenomenon emerges at the critical point. This first result is weaker than theone obtained by Talagrand (not limited to the average of the overlap), but weshow here that, at least on average, the overlap is proven to be non-negativewith no use of the Ghirlanda–Guerra identities. The latter are instead needed toobtain the second result, which is to control the behaviour of the overlap at thecritical point: we find the critical exponents of all the overlap correlation functions.