A note on the polynomial moments of the partition function in the SK model

We prove a simple identity relating the $k$th moment of the partition function $Z_N(\cdot)$ in the SK model to the $N$th moment of the partition function $Z_k(\cdot)$. As a corollary we find a characterisation of the limit $\lim_{N \to \infty} \frac{1}{N} \log \mathbb{E} Z_N(\beta)^k$ alternative to the one found previously by Michel Talagrand in \cite{4}.