  We have performed studies of the 3D random field $XY$ model on $L \times L
\times L$ simple cubic lattices with periodic boundary conditions, with a
random field strength of $h_r$ = 1.875, for $L =$ 64, 96 and 128, using a
parallelized Monte Carlo algorithm. We present results for the angle-averaged
magnetic structure factor, $S ( k )$ at $T$ = 1.00, which appears to be the
temperature at which small jumps in the magnetization per spin and the energy
per spin occur. The magnetization jump per spin scales with size roughly as
$L^{- 3/4}$, while the energy jump per spin scales like $L^{- 3/2}$. The
results also indicate the existence of an approximately logarithmic divergence
of $S ( k )$ as $k \to 0$. The magnetic susceptibility, $\chi (\vec{\bf k} = 0
)$, on the other hand, seems to have a value of about 14.2 under these
conditions. This suggests the absence of a ferromagnetic phase, and that the
lower critical dimension for long-range order in this model is three. Similar
results are found for $L$ = 64 samples at $h_r$ = 2.0 and $T$ = 0.875. We
expect that the behavior is qualitatively similar along the entire phase
boundary, but the scaling exponents may not be universal. These results appear
to be related to recent work on quantum disorder.
