---
res:
bibo_abstract:
- We study the unique solution m of the Dyson equation \( -m(z)^{-1} = z\1 - a +
S[m(z)] \) on a von Neumann algebra A with the constraint Imm≥0. Here, z lies
in the complex upper half-plane, a is a self-adjoint element of A and S is a positivity-preserving
linear operator on A. We show that m is the Stieltjes transform of a compactly
supported A-valued measure on R. Under suitable assumptions, we establish that
this measure has a uniformly 1/3-Hölder continuous density with respect to the
Lebesgue measure, which is supported on finitely many intervals, called bands.
In fact, the density is analytic inside the bands with a square-root growth at
the edges and internal cubic root cusps whenever the gap between two bands vanishes.
The shape of these singularities is universal and no other singularity may occur.
We give a precise asymptotic description of m near the singular points. These
asymptotics generalize the analysis at the regular edges given in the companion
paper on the Tracy-Widom universality for the edge eigenvalue statistics for correlated
random matrices [the first author et al., Ann. Probab. 48, No. 2, 963--1001 (2020;
Zbl 1434.60017)] and they play a key role in the proof of the Pearcey universality
at the cusp for Wigner-type matrices [G. Cipolloni et al., Pure Appl. Anal. 1,
No. 4, 615--707 (2019; Zbl 07142203); the second author et al., Commun. Math.
Phys. 378, No. 2, 1203--1278 (2020; Zbl 07236118)]. We also extend the finite
dimensional band mass formula from [the first author et al., loc. cit.] to the
von Neumann algebra setting by showing that the spectral mass of the bands is
topologically rigid under deformations and we conclude that these masses are quantized
in some important cases.@eng
bibo_authorlist:
- foaf_Person:
foaf_givenName: Johannes
foaf_name: Alt, Johannes
foaf_surname: Alt
foaf_workInfoHomepage: http://www.librecat.org/personId=36D3D8B6-F248-11E8-B48F-1D18A9856A87
- foaf_Person:
foaf_givenName: László
foaf_name: Erdös, László
foaf_surname: Erdös
foaf_workInfoHomepage: http://www.librecat.org/personId=4DBD5372-F248-11E8-B48F-1D18A9856A87
orcid: 0000-0001-5366-9603
- foaf_Person:
foaf_givenName: Torben H
foaf_name: Krüger, Torben H
foaf_surname: Krüger
foaf_workInfoHomepage: http://www.librecat.org/personId=3020C786-F248-11E8-B48F-1D18A9856A87
orcid: 0000-0002-4821-3297
bibo_doi: 10.4171/dm/780
bibo_volume: 25
dct_date: 2020^xs_gYear
dct_isPartOf:
- http://id.crossref.org/issn/1431-0635
- http://id.crossref.org/issn/1431-0643
dct_language: eng
dct_publisher: EMS Press@
dct_subject:
- General Mathematics
dct_title: 'The Dyson equation with linear self-energy: Spectral bands, edges and
cusps@'
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