  We present an analytical model for cosmological Lyman-limit systems (LLSs)
that successfully reproduces the observed evolution of the mean free path (L)
of ionizing photons. The evolution of the co-moving mean free path is
predominantly a consequence of the changing meta galactic photo-ionization rate
and the increase with cosmic time of the minimum mass below which halos lose
their gas due to photo-heating. In the model, Lyman-limit absorption is caused
by highly ionized gas in the outskirt of dark matter halos. We exploit the
association with halos to compute statistical properties of LLSs and of their
bias, b. The latter increases from 1.5 to 2.6 from redshifts 2 to 6. Combined
with the rapid increase with redshift of the bias of the halos that host a
quasar, the model predicts a rapid drop in the value of L when measured in
quasar spectra from z=5 to 6, whereas the actual value of L falls more
smoothly. We derive an expression for the effective optical depth due to Lyman
limit absorption as a function of wavelength and show that it depends
sensitively on the poorly constrained number density of LLSs as a function of
column density. The optical depth drops below unity for all wavelengths below a
redshift of 2.5, which is therefore the epoch when the Universe first became
transparent to ionizing photons.
