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.