  We analyse the spin wave approximation for the 2D-XY model, directly in
reciprocal space. In this limit the model is diagonal and the normal modes are
statistically independent. Despite this simplicity non-trivial critical
properties are observed and exploited. We confirm that the observed asymmetry
for the probability density function for order parameter fluctuations comes
from the divergence of the mode amplitudes across the Brillouin zone. We show
that the asymmetry is a many body effect despite the importance played by the
zone centre. The precise form of the function is dependent on the details of
the Gibbs measure, giving weight to the idea that an effective Gibbs measure
should exist in non-equilibrium systems, if a similar distribution is observed.
