  The study of random Fourier series, linear combinations of trigonometric
functions whose coefficients are independent (in our case Gaussian) random
variables with polynomially bounded means and standard deviations, dates back
to Norbert Wiener in one of the original constructions of Brownian motion. A
geometric generalization -- relevant e.g.\ to Euclidean quantum field theory
with an infrared cutoff -- is the study of random Gaussian linear combinations
of the eigenfunctions of the Laplace-Beltrami operator on an arbitrary compact
Riemannian manifold $(M,g)$, Gaussian noise.
  I will prove that, when our random coefficients are independent Gaussians
whose standard deviations obey polynomial asymptotics and whose means obey a
corresponding polynomial upper bound, the resultant random $\mathscr{H}^s
$-wavefront set (defined as a subset of the cosphere bundle $\mathbb{S}^*M$) is
either almost surely empty or almost surely the entirety of $\mathbb{S}^*M$,
depending on $s \in \mathbb{R}$, and we will compute the threshold $s$ and the
behavior of the wavefront set at it. The method of proof is as follows: using
Sazonov's theorem and its converse, it suffices to understand which
compositions of microlocal cutoffs and embeddings of $L^2$-based fractional
order Sobolev spaces are Hilbert-Schmidt (HS), and the answer follows from
general facts about the HS-norms of the elements of the pseudodifferential
calculus of Kohn and Nirenberg.
