Linear Amplifier Breakdown and Concentration Properties of a Gaussian Field Given that its $\bm{L^2}$-Norm is Large

TL;DR

This paper studies how Gaussian fields behave when their $L^2$-norm becomes large, showing they concentrate onto dominant eigenspaces and exploring implications for systems like Bose-Einstein condensation.

Contribution

It proves concentration properties of Gaussian fields with large $L^2$-norm, including in probability and mean, and discusses connections to physical phenomena such as Bose-Einstein condensation.

Findings

Concentration onto eigenspaces associated with the largest eigenvalue.

Probabilistic concentration in the trace class covariance case.

Mean concentration in the sup-norm for specific Gaussian fields.

Abstract

In the context of linear amplification for systems driven by the square of a Gaussian noise, we investigate the realizations of a Gaussian field in the limit where its $L^2$-norm is large. Concentration onto the eigenspace associated with the largest eigenvalue of the covariance of the field is proved. When the covariance is trace class, the concentration is in probability for the $L^2$-norm. A stronger concentration, in mean for the sup-norm, is proved for a smaller class of Gaussian fields, and an example of a field belonging to that class is given. A possible connection with Bose-Einstein condensation is briefly discussed.