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 foreground galactic halos at different transverse separations from background quasars represents a powerful tool to probe their gaseous environments. I compare predictions of state-of-the-art hydrodynamic cosmological simulations with the observed absorption around foreground quasars, damped Ly-a systems, and Lyman-break galaxies. I show that large-scale BOSS (Baryon Oscillation Spectroscopic Survey) and small-scale quasar pair measurements can be combined to precisely constrain the absorption profile over three decades in transverse distance (20kpc - 20Mpc). Far from galaxies (>2 Mpc), the simulations converge to the same profile and provide a reasonable match to the observations. This asymptotic agreement arises because the LCDM 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 - 2Mpc), illustrating the challenges of accurately modelling and resolving galaxy formation physics. It is noteworthy that these differences are observed as far out as ~2 Mpc, indicating that the sphere-of-influence of galaxies could extend to approximately ~20 times the halo virial radius (~100 kpc). Current observations are very precise on these scales and can thus strongly discriminate between different galaxy formation models. I 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.