TITLE: What is $\mathbb{E} [\max_{\sigma \in \{ \pm 1\}^n} \sigma^T Z \sigma]$ for a random Gaussian matrix $Z$?
QUESTION [5 upvotes]: Given an $n \times n$ random matrix $\mathbf{Z}$ with each entry i.i.d. $\mathcal{N} (0,1)$, what is $\mathbb{E} [\max_{\sigma \in \{ \pm 1\}^n} \sigma^T Z \sigma]$ as $n \to \infty$? If this is too much to ask, are there any good known upper or lower bounds?

REPLY [7 votes]: This is related to the Sherrington-Kirkpatrick Spin Glass model. In arxiv.org/pdf/1412.0170.pdf, Dmitry Panchenko writes that
$$\lim_{N\to\infty} \frac{1}{N} \mathbb{E}\big[\max_\sigma \frac{1}{\sqrt{N}} \sum_{i<j} g_{ij} \sigma_i \sigma_j \big] \approx 0.7633$$
with $g_{ij}$ iid standard normals. This is a consequence of the Parisi formula. I believe your desired asymptotic should then be $0.7633\sqrt{2}N^{3/2}+ o(N^{3/2})$.