  The XY model with quenched random disorder is studied by a zero temperature
domain wall renormalization group method in 2D and 3D. Instead of the usual
phase representation we use the charge (vortex) representation to compute the
domain wall, or defect, energy. For the gauge glass corresponding to the
maximum disorder we reconfirm earlier predictions that there is no ordered
phase in 2D but an ordered phase can exist in 3D at low temperature. However,
our simulations yield spin stiffness exponents $\theta_{s} \approx -0.36$ in 2D
and $\theta_{s} \approx +0.31$ in 3D, which are considerably larger than
previous estimates and strongly suggest that the lower critical dimension is
less than three. For the $\pm J$ XY spin glass in 3D, we obtain a spin
stiffness exponent $\theta_{s} \approx +0.10$ which supports the existence of
spin glass order at finite temperature in contrast with previous estimates
which obtain $\theta_{s}< 0$. Our method also allows us to study
renormalization group flows of both the coupling constant and the disorder
strength with length scale $L$. Our results are consistent with recent analytic
and numerical studies suggesting the absence of a re-entrant transition in 2D
at low temperature. Some possible consequences and connections with real vortex
systems are discussed.
