Open Access
2021 Operator level hard-to-soft transition for β-ensembles
Laure Dumaz, Yun Li, Benedek Valkó
Author Affiliations +
Electron. J. Probab. 26: 1-28 (2021). DOI: 10.1214/21-EJP602


The soft and hard edge scaling limits of β-ensembles can be characterized as the spectra of certain random Sturm-Liouville operators [12, 15]. It has been shown that by tuning the parameter of the hard edge process one can obtain the soft edge process as a scaling limit [3, 12, 14]. We prove that this limit can be realized on the level of the corresponding random operators. More precisely, the random operators can be coupled in a way so that the scaled versions of the hard edge operators converge to the soft edge operator a.s. in the norm resolvent sense.

Funding Statement

This research was partially supported by the ANR-16-CE93-0003 (Laure Dumaz) and the NSF award DMS-1712551 (Benedek Valkó).


The authors thank Cyril Labbé, Brian Rider and Bálint Virág for valuable discussions. LD and BV thank the hospitality of Centre International de Rencontres Mathématiques in Marseille where part of this work was originated.


Download Citation

Laure Dumaz. Yun Li. Benedek Valkó. "Operator level hard-to-soft transition for β-ensembles." Electron. J. Probab. 26 1 - 28, 2021.


Received: 24 March 2020; Accepted: 12 March 2021; Published: 2021
First available in Project Euclid: 7 April 2021

Digital Object Identifier: 10.1214/21-EJP602

Primary: 47B80 , 60B20
Secondary: 47E05

Keywords: random differential operators , random matrices

Vol.26 • 2021
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