Extremes of Gaussian fields with a product term in the variance

We study the high excursion probability of a centered Gaussian field on a square. Writing $\sigma$ and $r$ for its standard deviation and correlation function, we assume that $\sigma$ has a unique maximum at the corner $\boldsymbol{0}=(0,0)$ and $$1-\sigma(\boldsymbol{t}) \sim t_1^\beta+t_2^\beta+t_1^a t_2^a , \qquad \boldsymbol{t}=(t_1,t_2)\to\boldsymbol{0}$$ in $\mathbb R_+^2$. The local correlation is assumed to satisfy $$1-r(\boldsymbol{t},\boldsymbol{s})\sim |t_1-s_1|^\alpha+|t_2-s_2|^\alpha, \qquad 0<\alpha<\beta .$$ This product form of the standard-deviation loss is not covered by the usual locally additive assumptions. In the range $a<\beta/2$, the classical essential rectangle at the variance-loss scale no longer captures the leading contribution; the relevant localization becomes side-attached and, in one regime, effectively one-dimensional. We determine the corresponding high-level asymptotics, including the logarithmic and side-dominated regimes which do not arise in the locally additive case.