  Consider the classical XY model in a weak random external field pointing
along the $Y$ axis with strength $\epsilon$. We study the behavior of this
model as the range of the interaction is varied. We prove that in any dimension
$d \geq 2$ and for all $\epsilon$ sufficiently small, there is a range
$L=L(\epsilon)$ so that whenever the inverse temperature $\beta$ is larger than
some $\beta(\epsilon)$, there is strong residual ordering along the $X$
direction.
