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

Vincent Painchaud, Elliot Paquette

TL;DR

This work develops an operator-level transition from the soft edge Airyβ operator to the bulk sineβ operator within a unified canonical-systems framework for general β-ensembles. By embedding both operators as canonical systems and constructing a Brownian-coupling in a high-energy scaling, the authors prove vague convergence of the coefficient matrices, followed by convergence of transfer matrices, Weyl–Titchmarsh functions, and spectral measures, in probability. This yields the convergence of the rescaled Airyβ eigenvalue data to the sineβ point process, extending Valkó–Virág’s eigenvalue-process result to an operator-level setting and establishing a common framework for β-ensemble local limits. The approach emphasizes universality across β, connects stochastic Airy and sine operators, and provides tools to analyze spectral-data convergence via canonical-system methods. The results offer a robust pathway to understanding soft-edge to bulk transitions via a single, operator-theoretic lens with potential applicability to broader random-matrix limits.

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 $β$-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 Valkó and Virág (2009) by comparison to the Gaussian $β$-ensemble.

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

TL;DR

This work develops an operator-level transition from the soft edge Airyβ operator to the bulk sineβ operator within a unified canonical-systems framework for general β-ensembles. By embedding both operators as canonical systems and constructing a Brownian-coupling in a high-energy scaling, the authors prove vague convergence of the coefficient matrices, followed by convergence of transfer matrices, Weyl–Titchmarsh functions, and spectral measures, in probability. This yields the convergence of the rescaled Airyβ eigenvalue data to the sineβ point process, extending Valkó–Virág’s eigenvalue-process result to an operator-level setting and establishing a common framework for β-ensemble local limits. The approach emphasizes universality across β, connects stochastic Airy and sine operators, and provides tools to analyze spectral-data convergence via canonical-system methods. The results offer a robust pathway to understanding soft-edge to bulk transitions via a single, operator-theoretic lens with potential applicability to broader random-matrix limits.

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 -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 Valkó and Virág (2009) by comparison to the Gaussian -ensemble.

Paper Structure

This paper contains 40 sections, 39 theorems, 403 equations.

Table of Contents

  1. Introduction
  2. Airy and sine operators.
  3. Canonical system embedding.
  4. Related work on operator convergence.
  5. Related work on $\mathrm{Airy}_\beta$.
  6. Remarks on the point process convergence.
  7. Organization of the paper.
  8. Notation and conventions.
  9. Acknowledgements.
  10. Survey of canonical systems theory
  11. Canonical systems and their spectral theory
  12. Canonical systems and self-adjoint relations
  13. Self-adjoint realizations of the maximal relation
  14. Transfer matrices and Weyl--Titchmarsh functions
  15. Convergence of sequences of canonical systems
  16. ...and 25 more sections

Key Result

Theorem 1

Let $\mathcal{I} \coloneqq [0,1)$ if $\beta \leq 2$ and $\mathcal{I} \coloneqq [0,1]$ if $\beta > 2$. For any diverging sequence $\{E_n\}_{n\in\mathbb{N}} \subset (0,\infty)$, there are $\mathscr{C}^1$ bijections $\eta_{E_n}\colon [0,1+\varepsilon_{E_n}) \to [0,\infty)$ where $\varepsilon_{E_n} = 0$ in the vague topology of coefficient matrices on $\mathcal{I}$.

Theorems & Definitions (79)

  • Definition 1
  • Theorem 1
  • Remark
  • Corollary .1
  • Theorem 2
  • Remark
  • Theorem 3
  • Corollary \ref{thm.WTconvinlaw}.1
  • Definition 1
  • Theorem 4: Theorem 2.11 of remling_spectral_2018
  • ...and 69 more