Large deviations for discrete $β$-ensembles

We consider discrete $β$-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 their rightmost particle. We apply our general results to two classes of measures that are related to Jack symmetric functions.