On multiple peaks and moderate deviations for the supremum of a Gaussian field

We prove two theorems concerning extreme values of general Gaussian fields. Our first theorem concerns the phenomenon of multiple peaks. Consider a centered Gaussian field whose sites have variance at most $1$, and let $\rho$ be the standard deviation of its supremum. A theorem of Chatterjee states that when such a Gaussian field is superconcentrated (i.e., $\rho\ll1$), it typically attains values near its maximum on multiple almost-orthogonal sites and is said to exhibit multiple peaks. We improve his theorem in two respects: (i) the number of peaks attained by our bound is of the order $\exp(c/\rho^{2})$ (as opposed to Chatterjee’s polynomial bound in $1/\rho$) and (ii) our bound does not assume that the correlations are nonnegative. We also prove a similar result based on 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 almost-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.