Rigidity and Edge Universality of Discrete $β$-Ensembles

Alice Guionnet, Jiaoyang Huang

Published 2017-05-16Version 1

We study discrete $\beta$-ensembles as introduced in [17]. We obtain rigidity estimates on the particle locations, i.e. with high probability, the particles are close to their classical locations with an optimal error estimate. We prove the edge universality of the discrete $\beta$-ensemble, i.e. for $\beta\geq 1$, the distribution of extreme particles converges to the Tracy-Widom $\beta$ distribution. Our proof relies on Nekrasov's (or loop) equations, a multiscale analysis and a comparison argument with continuous $\beta$-ensembles.