  We prove two theorems concerning extreme values of general Gaussian fields.
Our first theorem concerns with the concept of multiple peaks. A theorem of
Chatterjee states that when a centered Gaussian field admits the so-called
superconcentration property, it typically attains values near its maximum on
multiple near-orthogonal sites, known as multiple peaks. We improve his theorem
in two aspects: (i) the number of peaks attained by our bound is of the order
$\exp(c / \sigma^2)$ (as opposed to Chatterjee's polynomial bound in
$1/\sigma$), where $\sigma$ is the standard deviation of the supremum of the
Gaussian field, which is assumed to have variance at most $1$ and (ii) our
bound need not assume that the correlations are non-negative. We also prove a
similar result based on the superconcentration of the free energy. As primary
applications, we infer that for the S-K spin glass model on the $n$-hypercube
and directed polymers on $\mathbb{Z}_n^2$, there are polynomially (in $n$) many
near-orthogonal sites that achieve values near their respective maxima.
  Our second theorem gives an upper bound on moderate deviation for the
supremum of a general Gaussian field. While the Gaussian isoperimetric
inequality implies a sub-Gaussian concentration bound for the supremum, we show
that the exponent in that bound can be improved under the assumption that the
expectation of the supremum is of the same order as that of the independent
case.
