  We consider discrete $\beta$-ensembles, as introduced by Borodin, Gorin and
Guionnet in (Publications math{\' e}matiques de l'IH{\' E}S 125, 1-78, 2017).
Under general assumptions, we establish a large deviation principle for the
empirical (or spectral) measures corresponding to these models. Our results
apply in the cases when the potential of the model depends on the number of
particles, and/or has slow growth near infinity, leading to an equilibrium
measure with infinite support.
