  We have performed studies of the 3D random field $XY$ model on 32 samples of
$L \times L \times L$ simple cubic lattices with periodic boundary conditions,
with a random field strength of $h_r$ = 1.5, for $L =$ 128, using a
parallelized Monte Carlo algorithm. We present results for the sample-averaged
magnetic structure factor, $S (\vec{\bf k})$ over a range of temperature, using
both random hot start and ferromagnetic cold start initial states, and
$\vec{\bf k}$ along the [1,0,0] and [1,1,1] directions. At $T =$ 1.875, $S
(\vec{\bf k})$ shows a broad peak near $|\vec{\bf k}| = 0$, with a correlation
length which is limited by thermal fluctuations, rather than the lattice size.
As $T$ is lowered, this peak grows and sharpens. By $T =$ 1.5, it is clear that
the correlation length is larger than $L =$ 128. The lowest temperature for
which $S (\vec{\bf k})$ was calculated is $T =$ 1.421875, where the hot start
and cold start initial conditions are usually not finding the same local
minimum in the phase space. Our results are consistent with the idea that there
is a finite value of $T$ below which $S (\vec{\bf k})$ diverges slowly as
$|\vec{\bf k}|$ goes to zero. This divergence would imply that the relaxation
time of the spins is also diverging. That is the signature of an
ergodicity-breaking phase transition.
