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. As part of the proof, we also obtain $t\to-\infty$ asymptotics for solutions to the eigenvalue equation of the stochastic Bessel operator.
From: Vincent Painchaud [view email]
[v1]
Tue, 7 Oct 2025 17:01:50 UTC (55 KB)
[v2]
Wed, 16 Sep 2026 16:49:59 UTC (59 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。