On the multiple overlap function of the SK model

Abstract

In this note, we prove an asymptotic expansion and a central limit theorem for the multiple overlap R1,...,s of the SK model, defined for given N, s ≥ 1 by R1,...,s = N-1 P i≤N σ 1 i . . . σs i. These results are obtained by a careful analysis of the terms appearing in the cavity derivation formula, as well as some graph induction procedures. Our method could hopefully be applied to other spin glasses models.