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