Extremes of Gaussian random fields with non-additive dependence structure

We derive exact asymptotics of $$mathbb{P}left(sup_{tin mathcal{A}}X(t)>uright), ~text{as}~ utoinfty,$$ for a centered Gaussian field $X(t),~tin mathcal{A}subsetmathbb{R}^n$, $n>1$ with continuous sample paths a.s. and general dependence structure, for which $arg max_{tin {mathcal{A}}} Var(X(t))$ is a Jordan set with finite and positive Lebesque measure of dimension $kleq n$. Our findings are applied to deriving the asymptotics of tail probabilities related to performance tables and dependent chi processes.