标题： 通过规范系统在$β$-系综中的算子层次硬边到整体过渡 显示英文标题

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

摘要： $\beta$-系综的硬边缘和整体缩放极限由随机贝塞尔算子和正弦算子描述，它们分别是一个随机的Sturm-Liouville算子和一个随机的Dirac算子。 通过将这两个算子表示为规范系统，我们证明在合适的高能缩放极限下，随机贝塞尔算子按分布收敛到随机正弦算子。 这首先在规范系统的系数矩阵的模糊拓扑中完成，然后扩展到相关Weyl-Titchmarsh函数和谱测度的收敛性。 证明依赖于驱动这两个算子的布朗运动之间的耦合，在此耦合下，收敛性在概率上成立。 显示英文摘要

摘要： 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.