  This paper explores the possibility of establishing an analytic form of the
distribution of the order parameter fluctuations in a two-dimensional critical
spin wave model, or width fluctuations of a two dimensional Edwards-Wilkinson
interface. It is shown that the characteristic function of the distribution can
be expressed exactly as a Gamma function quotient, while a Charlier series,
using the convolution of two Gumbel distributions as the kernel, converges to
the exact result over a restricted domain. These results can also be extended
to calculate the temperature dependence of the distribution and give an insight
into the origin of Gumbel-like distributions in steady-state and equilibrium
quantities that are not extreme values.
