---
_id: '149'
abstract:
- lang: eng
text: The eigenvalue density of many large random matrices is well approximated
by a deterministic measure, the self-consistent density of states. In the present
work, we show this behaviour for several classes of random matrices. In fact,
we establish that, in each of these classes, the self-consistent density of states
approximates the eigenvalue density of the random matrix on all scales slightly
above the typical eigenvalue spacing. For large classes of random matrices, the
self-consistent density of states exhibits several universal features. We prove
that, under suitable assumptions, random Gram matrices and Hermitian random matrices
with decaying correlations have a 1/3-Hölder continuous self-consistent density
of states ρ on R, which is analytic, where it is positive, and has either a square
root edge or a cubic root cusp, where it vanishes. We, thus, extend the validity
of the corresponding result for Wigner-type matrices from [4, 5, 7]. We show that
ρ is determined as the inverse Stieltjes transform of the normalized trace of
the unique solution m(z) to the Dyson equation −m(z) −1 = z − a + S[m(z)] on C
N×N with the constraint Im m(z) ≥ 0. Here, z lies in the complex upper half-plane,
a is a self-adjoint element of C N×N and S is a positivity-preserving operator
on C N×N encoding the first two moments of the random matrix. In order to analyze
a possible limit of ρ for N → ∞ and address some applications in free probability
theory, we also consider the Dyson equation on infinite dimensional von Neumann
algebras. We present two applications to random matrices. We first establish that,
under certain assumptions, large random matrices with independent entries have
a rotationally symmetric self-consistent density of states which is supported
on a centered disk in C. Moreover, it is infinitely often differentiable apart
from a jump on the boundary of this disk. Second, we show edge universality at
all regular (not necessarily extreme) spectral edges for Hermitian random matrices
with decaying correlations.
alternative_title:
- IST Austria Thesis
author:
- first_name: Johannes
full_name: Alt, Johannes
id: 36D3D8B6-F248-11E8-B48F-1D18A9856A87
last_name: Alt
citation:
ama: Alt J. *Dyson Equation and Eigenvalue Statistics of Random Matrices*.
IST Austria; 2018. doi:10.15479/AT:ISTA:TH_1040
apa: Alt, J. (2018). *Dyson equation and eigenvalue statistics of random matrices*.
IST Austria. https://doi.org/10.15479/AT:ISTA:TH_1040
chicago: Alt, Johannes. *Dyson Equation and Eigenvalue Statistics of Random Matrices*.
IST Austria, 2018. https://doi.org/10.15479/AT:ISTA:TH_1040.
ieee: J. Alt, *Dyson equation and eigenvalue statistics of random matrices*.
IST Austria, 2018.
ista: Alt J. 2018. Dyson equation and eigenvalue statistics of random matrices,
IST Austria, 456p.
mla: Alt, Johannes. *Dyson Equation and Eigenvalue Statistics of Random Matrices*.
IST Austria, 2018, doi:10.15479/AT:ISTA:TH_1040.
short: J. Alt, Dyson Equation and Eigenvalue Statistics of Random Matrices, IST
Austria, 2018.
date_created: 2018-12-11T11:44:53Z
date_published: 2018-07-12T00:00:00Z
date_updated: 2020-08-11T10:10:34Z
day: '12'
ddc:
- '515'
- '519'
department:
- _id: LaEr
doi: 10.15479/AT:ISTA:TH_1040
ec_funded: 1
file:
- access_level: open_access
checksum: d4dad55a7513f345706aaaba90cb1bb8
content_type: application/pdf
creator: dernst
date_created: 2019-04-08T13:55:20Z
date_updated: 2020-07-14T12:44:57Z
file_id: '6241'
file_name: 2018_thesis_Alt.pdf
file_size: 5801709
relation: main_file
- access_level: closed
checksum: d73fcf46300dce74c403f2b491148ab4
content_type: application/zip
creator: dernst
date_created: 2019-04-08T13:55:20Z
date_updated: 2020-07-14T12:44:57Z
file_id: '6242'
file_name: 2018_thesis_Alt_source.zip
file_size: 3802059
relation: source_file
file_date_updated: 2020-07-14T12:44:57Z
has_accepted_license: '1'
language:
- iso: eng
month: '07'
oa: 1
oa_version: Published Version
page: '456'
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '338804'
name: Random matrices, universality and disordered quantum systems
publication_status: published
publisher: IST Austria
publist_id: '7772'
pubrep_id: '1040'
related_material:
record:
- id: '6183'
relation: part_of_dissertation
status: public
- id: '6184'
relation: part_of_dissertation
status: public
- id: '1010'
relation: part_of_dissertation
status: public
- id: '1677'
relation: part_of_dissertation
status: public
- id: '550'
relation: part_of_dissertation
status: public
- id: '566'
relation: part_of_dissertation
status: public
- id: '6240'
relation: part_of_dissertation
status: public
status: public
supervisor:
- first_name: László
full_name: Erdös, László
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
title: Dyson equation and eigenvalue statistics of random matrices
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: dissertation
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2018'
...