Loop Equations Characterize Random Matrix Statistics

We prove that the universal local point processes of random matrix theory are characterized by their loop equation hierarchies. More precisely, for every rational $\beta >0$, the ${\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{e}}_{\beta }$ point process is the unique solution of the bulk loop equation hierarchy, and the ${\mathrm{A}\mathrm{i}\mathrm{r}\mathrm{y}}_{\beta }$ point process is the unique solution of the edge loop equation hierarchy. These uniqueness results provide a direct route to universality: it suffices to verify the corresponding approximate loop equations for the ensemble. In many models, these equations follow from local laws and integration by parts.