Asymptotics of Jack measures with homogeneous specializations

We consider Jack measures on partitions with homogeneous defining specializations. For each of the six distinct classes of measures obtained this way we prove a global law of large numbers with an explicit limiting particle density. We also demonstrate for one of these classes how to obtain a global central limit theorem, global and edge large deviation principles, and edge universality using the results of the paper. Our argument is based on explicitly evaluating Jack symmetric functions at homogeneous specializations, relating the Jack measures to the discrete $β$-ensembles from (Publications math{\' e}matiques de l'IH{\' E}S 125, 1-78, 2017) and using the discrete loop equations to substantially reduce computations.