On the multiple overlap function of the SK model

In this note, we prove an asymptotic expansion and a central limit theorem for the multiple overlap $R_{1,\dotsc,s}$ of the SK model, defined for given $N, s\ge 1$ by $R_{1,\dotsc, s}=N^{-1}\sum_{i\le N}\sigma_i^{1}\ldots\sigma_i^{ s}$. 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.