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

TL;DR

This paper demonstrates that in a high-energy limit, the stochastic Airy operator converges to the sine operator within a unified canonical systems framework, linking edge and bulk spectral limits of beta-ensembles.

Contribution

It establishes a convergence result between the Airy and sine operators as canonical systems, extending previous eigenvalue process results to operator level convergence.

Findings

Proves convergence of Airy to sine operator in the vague topology of canonical systems.

Shows convergence of spectral measures and Weyl-Titchmarsh functions.

Uses a coupling of Brownian paths to establish convergence in probability.

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.