We consider the random function \( H_N(\sigma) = \sum_{i,j=1}^{N} g_{ij} \sigma_i \sigma_j \) where the \( g_{ij} \) are i.i.d standard Gaussian and \( \sigma \in \{ \pm 1\}^N \).

\begin{equation} H_N(\sigma) = \frac{1}{\sqrt{N}} \sum_{i,j=1}^{N} g_{ij} \sigma_i \sigma_j \, , \end{equation}