  Quasar lines of sight intersect intervening galaxy discs or circum-galactic
environments at random impact parameters and potential well depths. Absorption
line velocity widths ($\Delta v_{90}$) are known to scale with host galaxy
stellar masses, and inversely with the projected separation from the quasar
line of sight. Its dependence on stellar mass can be eliminated by normalising
with the emission-line widths of the host galaxies, $\sigma_{em}$, so that
absorbers with a range of $\Delta v_{90}$ values can be compared directly.
Using a sample of DLA systems at 0.2 < z < 3.2 with spectroscopically confirmed
host galaxies, we find that the velocity ratio $\Delta v_{90}/\sigma_{em}$
decreases with projected distances from the hosts. We compare the data with
expectations of line-of-sight velocity dispersions derived for different dark
matter halo mass distributions, and find that models with steeper radial dark
matter profiles provide a better fit to the observations, although the scatter
remains large. Gas outflows from the galaxies may cause an increased scatter,
or scale radii of dark matter halo models may not be representative for the
galaxies. We demonstrate by computing virial velocities, that metal-rich DLAs
that belong to massive galaxy halos (M$_{halo} \approx 10^{12}$ M$_{\odot}$)
mostly remain gravitationally bound to the halos.
