  Galaxy formation depends critically on the physical state of gas in the
circumgalactic medium (CGM) and its interface with the intergalactic medium
(IGM), determined by the complex interplay between inflows from the IGM and
outflows from supernovae or AGN feedback. The average Lyman-alpha (Ly-a)
absorption profile around galactic halos represents a powerful tool to probe
their gaseous environments. We compare predictions from Illustris and Nyx
hydrodynamical simulations with the observed absorption around foreground
quasars, damped Ly-a systems, and Lyman-break galaxies. We show how large-scale
BOSS and small-scale quasar pair measurements can be combined to precisely
constrain the absorption profile over three decades in transverse distance
20kpc$\lesssim b\lesssim$20Mpc. Far from galaxies $\gtrsim2$Mpc, the
simulations converge to the same profile and provide a reasonable match to the
observations. This asymptotic agreement arises because the $\Lambda$CDM model
successfully describes the ambient IGM, and represents a critical advantage of
studying the mean absorption profile. However, significant differences between
the simulations, and between simulations and observations are present on scales
20kpc$\lesssim b\lesssim$2Mpc, illustrating the challenges of accurately
modeling and resolving galaxy formation physics. It is noteworthy that these
differences are observed as far out as $\sim2$Mpc, indicating that the
`sphere-of-influence' of galaxies could extend to approximately $\sim7$ times
the halo virial radius ($\sim100$kpc). Current observations are very precise on
these scales and can thus strongly discriminate between different galaxy
formation models. We demonstrate that the Ly-a absorption profile is primarily
sensitive to the underlying temperature-density relationship of diffuse gas
around galaxies, and argue that it thus provides a fundamental test of galaxy
formation models.
