We investigate how damped Lyman-$alpha$ absorbers (DLAs) at z ~ 2-3, detected in large optical spectroscopic surveys of quasars, trace the population of star-forming galaxies. Building on previous results, we construct a model based on observed and physically motivated scaling relations in order to reproduce the bivariate distributions of metallicity, Z, and HI column density, N(HI). Furthermore, the observed impact parameters for galaxies associated to DLAs are in agreement with the model predictions. The model strongly favours a metallicity gradient, which scales with the luminosity of the host galaxy, with a value of $gamma$* = -0.019 $pm$ 0.008 dex kpc$^{-1}$ for L* galaxies that gets steeper for fainter galaxies. We find that DLAs trace galaxies over a wide range of galaxy luminosities, however, the bulk of the DLA cross-section arises in galaxies with L ~ 0.1 L* at z ~ 2.5 broadly consistent with numerical simulations.
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