Exponential control of overlap in the replica method for p-spin Sherrington-Kirkpatrick model

Recently, Michel Talagrand computed the large deviations limit $\lim_{N\to\infty}(Na)^{-1}\log \e Z_N^a$ for the moments of the partition function $Z_N$ in the Sherrington-Kirkpatrick model for all real $a\geq 0.$ For $a\geq 1$ the limit is given by Guerra's inverse bound and this result extends the classical physicist's replica method that corresponds to integer $a.$ We give a new proof for $a\geq 1$ in the case of the pure $p$-spin SK model that provides a strong exponential control of the overlap.