---
_id: '6183'
abstract:
- lang: eng
text: "We study the unique solution $m$ of the Dyson equation \\[ -m(z)^{-1} = z
- a\r\n+ S[m(z)] \\] on a von Neumann algebra $\\mathcal{A}$ with the constraint\r\n$\\mathrm{Im}\\,m\\geq
0$. Here, $z$ lies in the complex upper half-plane, $a$ is\r\na self-adjoint element
of $\\mathcal{A}$ and $S$ is a positivity-preserving\r\nlinear operator on $\\mathcal{A}$.
We show that $m$ is the Stieltjes transform\r\nof a compactly supported $\\mathcal{A}$-valued
measure on $\\mathbb{R}$. Under\r\nsuitable assumptions, we establish that this
measure has a uniformly\r\n$1/3$-H\\\"{o}lder continuous density with respect
to the Lebesgue measure, which\r\nis supported on finitely many intervals, called
bands. In fact, the density is\r\nanalytic inside the bands with a square-root
growth at the edges and internal\r\ncubic root cusps whenever the gap between
two bands vanishes. The shape of\r\nthese singularities is universal and no other
singularity may occur. We give a\r\nprecise asymptotic description of $m$ near
the singular points. These\r\nasymptotics generalize the analysis at the regular
edges given in the companion\r\npaper on the Tracy-Widom universality for the
edge eigenvalue statistics for\r\ncorrelated random matrices [arXiv:1804.07744]
and they play a key role in the\r\nproof of the Pearcey universality at the cusp
for Wigner-type matrices\r\n[arXiv:1809.03971,arXiv:1811.04055]. We also extend
the finite dimensional band\r\nmass formula from [arXiv:1804.07744] to the von
Neumann algebra setting by\r\nshowing that the spectral mass of the bands is topologically
rigid under\r\ndeformations and we conclude that these masses are quantized in
some important\r\ncases."
article_number: '1804.07752'
article_processing_charge: No
author:
- first_name: Johannes
full_name: Alt, Johannes
id: 36D3D8B6-F248-11E8-B48F-1D18A9856A87
last_name: Alt
- first_name: László
full_name: Erdös, László
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
- first_name: Torben H
full_name: Krüger, Torben H
id: 3020C786-F248-11E8-B48F-1D18A9856A87
last_name: Krüger
orcid: 0000-0002-4821-3297
citation:
ama: 'Alt J, Erdös L, Krüger TH. The Dyson equation with linear self-energy: Spectral
bands, edges and cusps. *arXiv*.'
apa: 'Alt, J., Erdös, L., & Krüger, T. H. (n.d.). The Dyson equation with linear
self-energy: Spectral bands, edges and cusps. *arXiv*.'
chicago: 'Alt, Johannes, László Erdös, and Torben H Krüger. “The Dyson Equation
with Linear Self-Energy: Spectral Bands, Edges and Cusps.” *ArXiv*, n.d.'
ieee: 'J. Alt, L. Erdös, and T. H. Krüger, “The Dyson equation with linear self-energy:
Spectral bands, edges and cusps,” *arXiv*. .'
ista: 'Alt J, Erdös L, Krüger TH. The Dyson equation with linear self-energy: Spectral
bands, edges and cusps. arXiv, 1804.07752.'
mla: 'Alt, Johannes, et al. “The Dyson Equation with Linear Self-Energy: Spectral
Bands, Edges and Cusps.” *ArXiv*, 1804.07752.'
short: J. Alt, L. Erdös, T.H. Krüger, ArXiv (n.d.).
date_created: 2019-03-28T09:20:06Z
date_published: 2018-04-20T00:00:00Z
date_updated: 2021-01-12T08:06:36Z
day: '20'
department:
- _id: LaEr
external_id:
arxiv:
- '1804.07752'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1804.07752
month: '04'
oa: 1
oa_version: Preprint
publication: arXiv
publication_status: submitted
related_material:
record:
- id: '149'
relation: dissertation_contains
status: public
status: public
title: 'The Dyson equation with linear self-energy: Spectral bands, edges and cusps'
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2018'
...