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We derive, up to a constant factor, matching lower and upper bounds on theconcentration functions of suprema of separable centered Gaussian processes andorder statistics of Gaussian random fields. These bounds reveal that suprema ofseparable centered Gaussian processes $\{X_u : u \in U\}$ exhibit the sameanti-concentration properties as a single Gaussian random variable with meanzero and variance $\mathrm{Var}(\sup_{u \in U} X_u)$. To apply these results tohigh-dimensional statistical problems, it is therefore essential to understandthe asymptotic behavior of $\mathrm{Var}(\sup_{u \in U} X_u)$ as the dimensionor metric entropy of the index set $U$ increases. Consequently, we also derivelower and upper bounds on this quantity.

Anti-concentration of Suprema of Gaussian Processes and Gaussian Order Statistics