With close pairs of quasars at different redshifts, a background quasar sightline can be used to study a foreground quasars environment in absorption. We use a sample of 650 projected quasar pairs to study the HI Lya absorption transverse to luminous, z~2 quasars at proper separations of 30kpc < R < 1Mpc. In contrast to measurements along the line-of-sight, regions transverse to quasars exhibit enhanced HI Lya absorption and a larger variance than the ambient intergalactic medium, with increasing absorption and variance toward smaller scales. Analysis of composite spectra reveals excess absorption characterized by a Lya equivalent width profile W = 2.3A (R/100kpc)^-0.46. We also observe a high (~60%) covering factor of strong, optically thick HI absorbers (HI column log NHI > 17.3) at separations R<200kpc, which decreases to ~20% at R~1Mpc, but still represents a significant excess over the cosmic average. This excess of optically thick absorption can be described by a quasar-absorber cross-correlation function xi_QA(r) = (r/r_0)^gamma with a large correlation length r_0 = 12.5+2.7-1.4 Mpc/h (comoving) and gamma = 1.68+0.14-0.30. The HI absorption measured around quasars exceeds that of any previously studied population, consistent with quasars being hosted by massive dark matter halos Mhalo~10^12.5 Msun at z~2.5. The environments of these massive halos are highly biased towards producing optically thick gas, and may even dominate the cosmic abundance of Lyman limit systems and hence the intergalactic opacity to ionizing photons at z~2.5. The anisotropic absorption around quasars implies the transverse direction is much less likely to be illuminated by ionizing radiation than the line-of-sight, which we interpret in terms of the same obscuration effects frequently invoked in unified models of active galactic nuclei.
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