Universality of the Stochastic Bessel Operator

Abstract

We establish universality at the hard edge for general beta ensembles assuming that: the background potential V is a polynomial such that $$x \mapsto V(x^2)$$ is strongly convex, $$\beta \ge 1$$ , and the “dimension-difference” parameter $$a\ge 0$$ . The method rests on the corresponding tridiagonal matrix models, showing that their appropriate continuum scaling limit is given by the Stochastic Bessel Operator. As conjectured in Edelman and Sutton (J Stat Phys 127:1121–1165, 2007) and rigorously established in Ramirez and Rider (Commun Math Phys 288:887–906, 2009), the latter characterizes the hard edge in the case of linear potential and all $$\beta $$ (the classical “beta-Laguerre” ensembles).