  The recently discovered population of ultra-faint extended line emitters can
account for the majority of the incidence rate of Damped Lyman Alpha systems
(DLAs) at z ~ 3 if the line emission is interpreted as Ly alpha. We show here
that a model similar to that proposed by Haehnelt, Steinmetz, & Rauch (2000),
which explains the incidence rate and kinematics of DLAs in the context of
$\Lambda$CDM models for structure formation, also reproduces the size
distribution of the new population of faint Ly alpha emitters for plausible
parameters. This lends further support to identification of the emitters with
the hitherto elusive population of DLA host galaxies. The observed incidence
rate of DLAs together with the observed space density and size distribution of
the emitters suggest a duty cycle of ~ 0.2 - 0.4 for the Ly alpha emission from
DLA host galaxies. We further show that Ly alpha cooling is expected to
contribute little to the Ly alpha emission for the majority of emitters. This
leaves centrally concentrated star formation at a rate of a few tenths
M_sun/yr, surrounded by extended Ly alpha halos with radii up to 30-50 kpc, as
the most plausible explanation for the origin of the emission. Both the
luminosity function of Ly alpha emission and the velocity width distribution of
low ionization absorption require that galaxies inside Dark Matter (DM) halos
with virial velocities < 50 - 70 km/s contribute little to the incidence rate
of DLAs at z ~ 3, suggesting that energy and momentum input due to star
formation efficiently removes gas from these halos. Galaxies with DM halos with
virial velocities of 100 - 150 km/s appear to account for the majority of DLA
host galaxies. DLA host galaxies at z ~ 3 should thus become the building
blocks of typical present-day galaxies.
