Concentration properties of Gaussian random fields

TL;DR

This paper investigates how Gaussian random fields behave when conditioned on large quadratic forms, revealing they are dominated by principal eigenmodes, with applications to high maxima and flow structures.

Contribution

It establishes that Gaussian fields conditioned on large quadratic forms are governed by principal eigenmodes, extending previous results and applying to flow structures with high helicity.

Findings

Gaussian fields conditioned on large quadratic forms align with principal eigenmodes.

Re-derivation of Adler's result on high local maxima structure.

Application to Gaussian flow reveals flow structure at high helicity.

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.