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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.

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  • $\begingroup$ Will you point to some of the literature you mention? I am confused because the question asks about a process, which usually means something that evolves, but there is no reference to time in the question. $\endgroup$
    – user44143
    Aug 20, 2022 at 11:51
  • $\begingroup$ Sure, 'On maximum of Gaussian random field having unique maximum point of its variance' by Kobelkov and Piterbarg contains some bounds, that can be applied. They presume the index set to be the closure of an open set in $\mathbb{R}^d$, which our setting can be also written as with open set $B^{\mathbb{R}^d}_1(0)$. $\endgroup$
    – Tardis
    Aug 20, 2022 at 12:58
  • $\begingroup$ Also the following paper I found since asking the question seems like the right setting, but I'm not sure, if my process has a version with $C^2$-paths. 'A general expression for the distribution of the maximum of a Gaussian field and the approximation of the tail' by Azais and Wschebor $\endgroup$
    – Tardis
    Aug 20, 2022 at 13:01
  • $\begingroup$ Sorry, I'm used to the general definition where a process is just a family of random variables. A random field would be more accurate here. $\endgroup$
    – Tardis
    Aug 20, 2022 at 13:03

1 Answer 1

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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.

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