Universality of the Stochastic Bessel Operator

TL;DR

This paper proves the universal behavior at the hard edge for a broad class of beta ensembles with polynomial potentials, showing their scaled limits are described by the Stochastic Bessel Operator.

Contribution

It generalizes the universality of the Stochastic Bessel Operator to polynomial potentials with convexity conditions for all beta ≥ 1, extending previous results beyond linear cases.

Findings

Universal hard edge behavior established for general beta ensembles.

Continuum limits of tridiagonal models are given by the Stochastic Bessel Operator.

Results include classical beta-Laguerre ensembles as special cases.

Abstract

We establish universality at the hard edge for general beta ensembles provided that the background potential V is a polynomial such that x -> V(x^2) is uniformly convex and beta is larger than or equal to one. 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 by Edelman-Sutton and rigorously established by Ramirez-Rider, the latter characterizes the hard edge in the case of linear potential and all beta (the classical "beta-Laguerre" ensembles)