Let $S\_{d-1}$ denote the unit sphere in $\mathbb{R}^d$ and let $(Z\_x)\_{x \in S\_{d-1}}$ be a gaussian process with mean zero and covariance structure given by the square of the scalar product, i.e.
$$
\operatorname{Cov}[Z\_x, Z\_y] = \langle x,y \rangle^2 = (x \cdot y)^2 \ .
$$
I am interested in the distribution of $\max\limits\_{x \in S\_{d-1}} Z\_x$ and most of the literature I can find gives only bounds. Can someone point me to some relevant literature or give me some tipps on how to describe the distribution of $\max\limits\_{x \in S\_{d-1}} Z\_x$?


Any help is much apprechiated.



Maximum of a certain Gaussian field
Your field is the $2$-spin, that is can be represented as $Z\_x=\sum\_{i,j} J\_{ij} x\_i x\_j$, where $J\_{i,j} $ are iid Gaussian (up to symmetry, ie $J\_{i,j}=J\_{j,i}$ and the diagonal has twice the variance of off-diagonal. In short, $J$ is a GOE matrix. In particular, the maximum you seek is the top eigenvalue of $J$, whose distribution is well known for $d$ large.


