Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Operator level hard-to-soft transition for $\beta$-ensembles   The soft and hard edge scaling limits of $\beta$-ensembles can be
characterized as the spectra of certain random Sturm-Liouville operators. 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. 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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The categories of this paper are: cs
32
The categories of this paper are: atom-ph
21
The categories of this paper are: adap-org
11
The categories of this paper are: hep-lat
33
The categories of this paper are: plasm-ph
5
The categories of this paper are: gr-qc
37
The categories of this paper are: supr-con
3
The categories of this paper are: physics
17
The categories of this paper are: q-fin
22
The categories of this paper are: chao-dyn
31
The categories of this paper are: ao-sci
36
The categories of this paper are: cmp-lg
15
The categories of this paper are: nucl-ex
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The categories of this paper are: hep-ex
18
The categories of this paper are: econ
13
The categories of this paper are: quant-ph