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.
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.
"Operator level hard-to-soft transition for β-ensembles." Electron. J. Probab. 26 1 - 28, 2021. https://doi.org/10.1214/21-EJP602