Concentration properties of Gaussian random fields

TL;DR

It is proved that in such a case the Gaussian field is primarily governed by the fundamental eigenmode of a particular operator, given that a particular real quadratic form $\mathcal{Q}$ is arbitrarily large.

Abstract

We study the problem of a random Gaussian vector field given that a particular real quadratic form $\mathcal{Q}$ is arbitrarily large. We prove that in such a case the Gaussian field is primarily governed by the fundamental eigenmode of a particular operator. As a good check of our proposition we use it to re-derive the result of Adler dealing with the structure of field in the vicinity of a high local maxima. We have also applied our result to an incompressible homogeneous Gaussian random flow in the limit of large local helicity and calculate the structure of the flow.