Operator level hard edge to bulk transition in $β$-ensembles via canonical systems

Abstract: The hard edge and bulk scaling limits of $\beta$-ensembles are described by the stochastic Bessel and sine operators, which are respectively a random Sturm-Liouville operator and a random Dirac operator. By representing both operators as canonical systems, we show that in a suitable high-energy scaling limit, the stochastic Bessel operator converges in law to the stochastic sine operator. This is first done in the vague topology of canonical systems' coefficient matrices, and then extended to the convergence of the associated Weyl-Titchmarsh functions and spectral measures. The proof relies on a coupling between the Brownian motions that drive the two operators, under which the convergence holds in probability.