  We perform a semi-automated survey for tau>=2 Lyman Limit systems (LLSs) in
quasar spectra from the Sloan Digital Sky Survey, Data Release 7. From a
starting sample of 2473 quasars with zem=3.6-4.4, we analyze 469 spectra
meeting strict seletion criteria for a total redshift path Dz=93.8 and identify
192 intervening systems at z>3.3. The incidence of tau>=2 LLSs per unit
redshift, l(z), is well described by a single-power law at these redshifts:
l(z) = C_LLS [(1+z)/(1+z_*)]^gamma, with z_*=3.7, C_LLS = 1.9+/-0.2, and gamma
= 5.2+/-1.5 (68% c.l.). These values are systematically lower than previous
estimates (especially at z<4) but are consistent with recent measurements of
the mean free path to ionizing radiation. Extrapolations of this power-law to
z=0 are inconsistent with previous estimations of l(z) at z<1 and suggest a
break at z~2, similar to that observed for the Lya forest. Our results also
indicate that the systems giving rise to LLS absorption decrease by ~50% in
comoving number density and/or physical size from z=4 to 3.3, perhaps due to an
enhanced extragalactic ultraviolet background. The observations place an
integral constraint on the HI frequency distribution f(N_HI,X) and indicate
that the power-law slope beta= dln[f(N,X)]/dln[N] is likely shallower than beta
= -1 at N_HI=10^18 cm^-2. Including other constraints on f(N_HI,X) from the
literature, we infer that beta is steeper than beta = -1.7 at N_HI~10^15 cm^-2,
implying at least two inflections in f(N_HI,X). We also perform a survey for
proximate LLSs (PLLSs) and find that l(z)_PLLS is systematically lower ~25%
than intervening systems. Finally, we estimate that systematic effects impose
an uncertainty of 10-20% in the l(z) measurements; these effects may limit the
precision of all future surveys.
