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

Abstract: The stochastic Airy and sine operators, which are respectively a random Sturm-Liouville operator and a random Dirac operator, characterize the soft edge and bulk scaling limits of $\beta$-ensembles. Dirac and Sturm-Liouville operators are distinct operator classes which can both be represented as canonical systems, which gives a unified framework for defining important properties, such as their spectral data. Seeing both as canonical systems, we prove that in a suitable high-energy scaling limit, the Airy operator converges to the sine operator. We prove this convergence in the vague topology of canonical systems' coefficient matrices, and deduce the convergence of the associated Weyl-Titchmarsh functions and spectral measures. Our proof relies on a coupling between the Brownian paths that drive the two operators, under which the convergence holds in probability. This extends the corresponding result at the level of the eigenvalue point processes, proven by Valk\'o and Vir\'ag (2009) by comparison to the Gaussian $\beta$-ensemble.