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 $\mathbf{0}=(0,0)$ and $$1-\sigma (\mathit{t})\sim t_1^{\beta }+t_2^{\beta }+t_1^a{t}_2^a,\mathit{t}=(t_1,t_2)\to \mathbf{0}$$ in ${\mathbb{R}}_{+}^2$. The local correlation is assumed to satisfy $$1-r(\mathit{t},\mathit{s})\sim \mathrm{\mid }{t}_1-s_1{\mathrm{\mid }}^{\alpha }+\mathrm{\mid }{t}_2-s_2{\mathrm{\mid }}^{\alpha },0<\alpha <\beta \mathrm{.}$$ This product form of the standard-deviation loss is not covered by the usual locally additive assumptions. In the range $a<\beta \mathrm{/}2$, 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.