  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.
