Abstract
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.
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