  We examine the kinematics of the gas in the environments of galaxies hosting
quasars at $z\sim2$. We employ 148 projected quasar pairs to study the
circumgalactic gas of the foreground quasars in absorption. The sample selects
foreground quasars with precise redshift measurements, using emission-lines
with precision $\lesssim300\,{\rm km\,s^{-1}}$ and average offsets from the
systemic redshift $\lesssim|100\,{\rm km\,s^{-1}}|$. We stack the background
quasar spectra at the foreground quasar's systemic redshift to study the mean
absorption in \ion{C}{2}, \ion{C}{4}, and \ion{Mg}{2}. We find that the mean
absorptions exhibit large velocity widths $\sigma_v\approx300\,{\rm
km\,s^{-1}}$. Further, the mean absorptions appear to be asymmetric about the
systemic redshifts. The mean absorption centroids exhibit small redshift
relative to the systemic $\delta v\approx+200\,{\rm km\,s^{-1}}$, with large
intrinsic scatter in the centroid velocities of the individual absorption
systems. We find the observed widths are consistent with gas in gravitational
motion and Hubble flow. However, while the observation of large widths alone
does not require galactic-scale outflows, the observed offsets suggest that the
gas is on average outflowing from the galaxy. The observed offsets also suggest
that the ionizing radiation from the foreground quasars is anisotropic and/or
intermittent.
