---
_id: '14694'
abstract:
- lang: eng
text: 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.
article_processing_charge: Yes
article_type: original
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. Documenta Mathematica. 2020;25:1421-1539. doi:10.4171/dm/780'
apa: 'Alt, J., Erdös, L., & Krüger, T. H. (2020). The Dyson equation with linear
self-energy: Spectral bands, edges and cusps. Documenta Mathematica. EMS
Press. https://doi.org/10.4171/dm/780'
chicago: 'Alt, Johannes, László Erdös, and Torben H Krüger. “The Dyson Equation
with Linear Self-Energy: Spectral Bands, Edges and Cusps.” Documenta Mathematica.
EMS Press, 2020. https://doi.org/10.4171/dm/780.'
ieee: 'J. Alt, L. Erdös, and T. H. Krüger, “The Dyson equation with linear self-energy:
Spectral bands, edges and cusps,” Documenta Mathematica, vol. 25. EMS Press,
pp. 1421–1539, 2020.'
ista: 'Alt J, Erdös L, Krüger TH. 2020. The Dyson equation with linear self-energy:
Spectral bands, edges and cusps. Documenta Mathematica. 25, 1421–1539.'
mla: 'Alt, Johannes, et al. “The Dyson Equation with Linear Self-Energy: Spectral
Bands, Edges and Cusps.” Documenta Mathematica, vol. 25, EMS Press, 2020,
pp. 1421–539, doi:10.4171/dm/780.'
short: J. Alt, L. Erdös, T.H. Krüger, Documenta Mathematica 25 (2020) 1421–1539.
date_created: 2023-12-18T10:37:43Z
date_published: 2020-09-01T00:00:00Z
date_updated: 2023-12-18T10:46:09Z
day: '01'
ddc:
- '510'
department:
- _id: LaEr
doi: 10.4171/dm/780
external_id:
arxiv:
- '1804.07752'
file:
- access_level: open_access
checksum: 12aacc1d63b852ff9a51c1f6b218d4a6
content_type: application/pdf
creator: dernst
date_created: 2023-12-18T10:42:32Z
date_updated: 2023-12-18T10:42:32Z
file_id: '14695'
file_name: 2020_DocumentaMathematica_Alt.pdf
file_size: 1374708
relation: main_file
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file_date_updated: 2023-12-18T10:42:32Z
has_accepted_license: '1'
intvolume: ' 25'
keyword:
- General Mathematics
language:
- iso: eng
month: '09'
oa: 1
oa_version: Published Version
page: 1421-1539
publication: Documenta Mathematica
publication_identifier:
eissn:
- 1431-0643
issn:
- 1431-0635
publication_status: published
publisher: EMS Press
quality_controlled: '1'
related_material:
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- id: '6183'
relation: earlier_version
status: public
status: public
title: 'The Dyson equation with linear self-energy: Spectral bands, edges and cusps'
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: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 25
year: '2020'
...
---
_id: '6184'
abstract:
- lang: eng
text: We prove edge universality for a general class of correlated real symmetric
or complex Hermitian Wigner matrices with arbitrary expectation. Our theorem also
applies to internal edges of the self-consistent density of states. In particular,
we establish a strong form of band rigidity which excludes mismatches between
location and label of eigenvalues close to internal edges in these general models.
article_processing_charge: No
article_type: original
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
- first_name: Dominik J
full_name: Schröder, Dominik J
id: 408ED176-F248-11E8-B48F-1D18A9856A87
last_name: Schröder
orcid: 0000-0002-2904-1856
citation:
ama: 'Alt J, Erdös L, Krüger TH, Schröder DJ. Correlated random matrices: Band rigidity
and edge universality. Annals of Probability. 2020;48(2):963-1001. doi:10.1214/19-AOP1379'
apa: 'Alt, J., Erdös, L., Krüger, T. H., & Schröder, D. J. (2020). Correlated
random matrices: Band rigidity and edge universality. Annals of Probability.
Institute of Mathematical Statistics. https://doi.org/10.1214/19-AOP1379'
chicago: 'Alt, Johannes, László Erdös, Torben H Krüger, and Dominik J Schröder.
“Correlated Random Matrices: Band Rigidity and Edge Universality.” Annals of
Probability. Institute of Mathematical Statistics, 2020. https://doi.org/10.1214/19-AOP1379.'
ieee: 'J. Alt, L. Erdös, T. H. Krüger, and D. J. Schröder, “Correlated random matrices:
Band rigidity and edge universality,” Annals of Probability, vol. 48, no.
2. Institute of Mathematical Statistics, pp. 963–1001, 2020.'
ista: 'Alt J, Erdös L, Krüger TH, Schröder DJ. 2020. Correlated random matrices:
Band rigidity and edge universality. Annals of Probability. 48(2), 963–1001.'
mla: 'Alt, Johannes, et al. “Correlated Random Matrices: Band Rigidity and Edge
Universality.” Annals of Probability, vol. 48, no. 2, Institute of Mathematical
Statistics, 2020, pp. 963–1001, doi:10.1214/19-AOP1379.'
short: J. Alt, L. Erdös, T.H. Krüger, D.J. Schröder, Annals of Probability 48 (2020)
963–1001.
date_created: 2019-03-28T09:20:08Z
date_published: 2020-03-01T00:00:00Z
date_updated: 2024-02-22T14:34:33Z
day: '01'
department:
- _id: LaEr
doi: 10.1214/19-AOP1379
ec_funded: 1
external_id:
arxiv:
- '1804.07744'
isi:
- '000528269100013'
intvolume: ' 48'
isi: 1
issue: '2'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1804.07744
month: '03'
oa: 1
oa_version: Preprint
page: 963-1001
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '338804'
name: Random matrices, universality and disordered quantum systems
publication: Annals of Probability
publication_identifier:
issn:
- 0091-1798
publication_status: published
publisher: Institute of Mathematical Statistics
quality_controlled: '1'
related_material:
record:
- id: '149'
relation: dissertation_contains
status: public
- id: '6179'
relation: dissertation_contains
status: public
scopus_import: '1'
status: public
title: 'Correlated random matrices: Band rigidity and edge universality'
type: journal_article
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 48
year: '2020'
...
---
_id: '15063'
abstract:
- lang: eng
text: We consider the least singular value of a large random matrix with real or
complex i.i.d. Gaussian entries shifted by a constant z∈C. We prove an optimal
lower tail estimate on this singular value in the critical regime where z is around
the spectral edge, thus improving the classical bound of Sankar, Spielman and
Teng (SIAM J. Matrix Anal. Appl. 28:2 (2006), 446–476) for the particular shift-perturbation
in the edge regime. Lacking Brézin–Hikami formulas in the real case, we rely on
the superbosonization formula (Comm. Math. Phys. 283:2 (2008), 343–395).
acknowledgement: Partially supported by ERC Advanced Grant No. 338804. This project
has received funding from the European Union’s Horizon 2020 research and innovation
programme under the Marie Sklodowska-Curie Grant Agreement No. 66538
article_processing_charge: No
article_type: original
author:
- first_name: Giorgio
full_name: Cipolloni, Giorgio
id: 42198EFA-F248-11E8-B48F-1D18A9856A87
last_name: Cipolloni
orcid: 0000-0002-4901-7992
- first_name: László
full_name: Erdös, László
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
- first_name: Dominik J
full_name: Schröder, Dominik J
id: 408ED176-F248-11E8-B48F-1D18A9856A87
last_name: Schröder
orcid: 0000-0002-2904-1856
citation:
ama: Cipolloni G, Erdös L, Schröder DJ. Optimal lower bound on the least singular
value of the shifted Ginibre ensemble. Probability and Mathematical Physics.
2020;1(1):101-146. doi:10.2140/pmp.2020.1.101
apa: Cipolloni, G., Erdös, L., & Schröder, D. J. (2020). Optimal lower bound
on the least singular value of the shifted Ginibre ensemble. Probability and
Mathematical Physics. Mathematical Sciences Publishers. https://doi.org/10.2140/pmp.2020.1.101
chicago: Cipolloni, Giorgio, László Erdös, and Dominik J Schröder. “Optimal Lower
Bound on the Least Singular Value of the Shifted Ginibre Ensemble.” Probability
and Mathematical Physics. Mathematical Sciences Publishers, 2020. https://doi.org/10.2140/pmp.2020.1.101.
ieee: G. Cipolloni, L. Erdös, and D. J. Schröder, “Optimal lower bound on the least
singular value of the shifted Ginibre ensemble,” Probability and Mathematical
Physics, vol. 1, no. 1. Mathematical Sciences Publishers, pp. 101–146, 2020.
ista: Cipolloni G, Erdös L, Schröder DJ. 2020. Optimal lower bound on the least
singular value of the shifted Ginibre ensemble. Probability and Mathematical Physics.
1(1), 101–146.
mla: Cipolloni, Giorgio, et al. “Optimal Lower Bound on the Least Singular Value
of the Shifted Ginibre Ensemble.” Probability and Mathematical Physics,
vol. 1, no. 1, Mathematical Sciences Publishers, 2020, pp. 101–46, doi:10.2140/pmp.2020.1.101.
short: G. Cipolloni, L. Erdös, D.J. Schröder, Probability and Mathematical Physics
1 (2020) 101–146.
date_created: 2024-03-04T10:27:57Z
date_published: 2020-11-16T00:00:00Z
date_updated: 2024-03-04T10:33:15Z
day: '16'
department:
- _id: LaEr
doi: 10.2140/pmp.2020.1.101
ec_funded: 1
external_id:
arxiv:
- '1908.01653'
intvolume: ' 1'
issue: '1'
keyword:
- General Medicine
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.1908.01653
month: '11'
oa: 1
oa_version: Preprint
page: 101-146
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '338804'
name: Random matrices, universality and disordered quantum systems
- _id: 2564DBCA-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '665385'
name: International IST Doctoral Program
publication: Probability and Mathematical Physics
publication_identifier:
issn:
- 2690-1005
- 2690-0998
publication_status: published
publisher: Mathematical Sciences Publishers
quality_controlled: '1'
scopus_import: '1'
status: public
title: Optimal lower bound on the least singular value of the shifted Ginibre ensemble
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 1
year: '2020'
...
---
_id: '15079'
abstract:
- lang: eng
text: "Large complex systems tend to develop universal patterns that often represent
their essential characteristics. For example, the cumulative effects of independent
or weakly dependent random variables often yield the Gaussian universality class
via the central limit theorem. For non-commutative random variables, e.g. matrices,
the Gaussian behavior is often replaced by another universality class, commonly
called random matrix statistics. Nearby eigenvalues are strongly correlated, and,
remarkably, their correlation structure is universal, depending only on the symmetry
type of the matrix. Even more surprisingly, this feature is not restricted to
matrices; in fact Eugene Wigner, the pioneer of the field, discovered in the 1950s
that distributions of the gaps between energy levels of complicated quantum systems
universally follow the same random matrix statistics. This claim has never been
rigorously proved for any realistic physical system but experimental data and
extensive numerics leave no doubt as to its correctness. Since then random matrices
have proved to be extremely useful phenomenological models in a wide range of
applications beyond quantum physics that include number theory, statistics, neuroscience,
population dynamics, wireless communication and mathematical finance. The ubiquity
of random matrices in natural sciences is still a mystery, but recent years have
witnessed a breakthrough in the mathematical description of the statistical structure
of their spectrum. Random matrices and closely related areas such as log-gases
have become an extremely active research area in probability theory.\r\nThis workshop
brought together outstanding researchers from a variety of mathematical backgrounds
whose areas of research are linked to random matrices. While there are strong
links between their motivations, the techniques used by these researchers span
a large swath of mathematics, ranging from purely algebraic techniques to stochastic
analysis, classical probability theory, operator algebra, supersymmetry, orthogonal
polynomials, etc."
article_processing_charge: No
article_type: original
author:
- first_name: László
full_name: Erdös, László
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
- first_name: Friedrich
full_name: Götze, Friedrich
last_name: Götze
- first_name: Alice
full_name: Guionnet, Alice
last_name: Guionnet
citation:
ama: Erdös L, Götze F, Guionnet A. Random matrices. Oberwolfach Reports.
2020;16(4):3459-3527. doi:10.4171/owr/2019/56
apa: Erdös, L., Götze, F., & Guionnet, A. (2020). Random matrices. Oberwolfach
Reports. European Mathematical Society. https://doi.org/10.4171/owr/2019/56
chicago: Erdös, László, Friedrich Götze, and Alice Guionnet. “Random Matrices.”
Oberwolfach Reports. European Mathematical Society, 2020. https://doi.org/10.4171/owr/2019/56.
ieee: L. Erdös, F. Götze, and A. Guionnet, “Random matrices,” Oberwolfach Reports,
vol. 16, no. 4. European Mathematical Society, pp. 3459–3527, 2020.
ista: Erdös L, Götze F, Guionnet A. 2020. Random matrices. Oberwolfach Reports.
16(4), 3459–3527.
mla: Erdös, László, et al. “Random Matrices.” Oberwolfach Reports, vol. 16,
no. 4, European Mathematical Society, 2020, pp. 3459–527, doi:10.4171/owr/2019/56.
short: L. Erdös, F. Götze, A. Guionnet, Oberwolfach Reports 16 (2020) 3459–3527.
date_created: 2024-03-05T07:54:44Z
date_published: 2020-11-19T00:00:00Z
date_updated: 2024-03-12T12:25:18Z
day: '19'
department:
- _id: LaEr
doi: 10.4171/owr/2019/56
intvolume: ' 16'
issue: '4'
language:
- iso: eng
month: '11'
oa_version: None
page: 3459-3527
publication: Oberwolfach Reports
publication_identifier:
issn:
- 1660-8933
publication_status: published
publisher: European Mathematical Society
quality_controlled: '1'
status: public
title: Random matrices
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 16
year: '2020'
...
---
_id: '7035'
abstract:
- lang: eng
text: 'The aim of this short note is to expound one particular issue that was discussed
during the talk [10] given at the symposium ”Researches on isometries as preserver
problems and related topics” at Kyoto RIMS. That is, the role of Dirac masses
by describing the isometry group of various metric spaces of probability measures. This article is of survey character, and it does not contain any essentially new
results.From an isometric point of view, in some cases, metric spaces of measures
are similar to C(K)-type function spaces. Similarity means here that their isometries are driven by some nice transformations
of the underlying space. Of course, it depends on the particular choice of the metric how nice these
transformations should be. Sometimes, as we will see, being a homeomorphism is
enough to generate an isometry. But sometimes we need more: the transformation
must preserve the underlying distance as well. Statements claiming that isometries
in questions are necessarily induced by homeomorphisms are called Banach-Stone-type
results, while results asserting that the underlying transformation is necessarily
an isometry are termed as isometric rigidity results.As Dirac masses can be considered as building bricks of the set of all Borel measures, a natural
question arises:Is it enough to understand how an isometry acts on the set of
Dirac masses? Does this action extend uniquely to all measures?In what follows,
we will thoroughly investigate this question.'
article_processing_charge: No
author:
- first_name: Gyorgy Pal
full_name: Geher, Gyorgy Pal
last_name: Geher
- first_name: Tamas
full_name: Titkos, Tamas
last_name: Titkos
- first_name: Daniel
full_name: Virosztek, Daniel
id: 48DB45DA-F248-11E8-B48F-1D18A9856A87
last_name: Virosztek
orcid: 0000-0003-1109-5511
citation:
ama: 'Geher GP, Titkos T, Virosztek D. Dirac masses and isometric rigidity. In:
Kyoto RIMS Kôkyûroku. Vol 2125. Research Institute for Mathematical Sciences,
Kyoto University; 2019:34-41.'
apa: 'Geher, G. P., Titkos, T., & Virosztek, D. (2019). Dirac masses and isometric
rigidity. In Kyoto RIMS Kôkyûroku (Vol. 2125, pp. 34–41). Kyoto, Japan:
Research Institute for Mathematical Sciences, Kyoto University.'
chicago: Geher, Gyorgy Pal, Tamas Titkos, and Daniel Virosztek. “Dirac Masses and
Isometric Rigidity.” In Kyoto RIMS Kôkyûroku, 2125:34–41. Research Institute
for Mathematical Sciences, Kyoto University, 2019.
ieee: G. P. Geher, T. Titkos, and D. Virosztek, “Dirac masses and isometric rigidity,”
in Kyoto RIMS Kôkyûroku, Kyoto, Japan, 2019, vol. 2125, pp. 34–41.
ista: Geher GP, Titkos T, Virosztek D. 2019. Dirac masses and isometric rigidity.
Kyoto RIMS Kôkyûroku. Research on isometries as preserver problems and related
topics vol. 2125, 34–41.
mla: Geher, Gyorgy Pal, et al. “Dirac Masses and Isometric Rigidity.” Kyoto RIMS
Kôkyûroku, vol. 2125, Research Institute for Mathematical Sciences, Kyoto
University, 2019, pp. 34–41.
short: G.P. Geher, T. Titkos, D. Virosztek, in:, Kyoto RIMS Kôkyûroku, Research
Institute for Mathematical Sciences, Kyoto University, 2019, pp. 34–41.
conference:
end_date: 2019-01-30
location: Kyoto, Japan
name: Research on isometries as preserver problems and related topics
start_date: 2019-01-28
date_created: 2019-11-18T15:39:53Z
date_published: 2019-01-30T00:00:00Z
date_updated: 2021-01-12T08:11:33Z
day: '30'
department:
- _id: LaEr
intvolume: ' 2125'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/2125.html
month: '01'
oa: 1
oa_version: Submitted Version
page: 34-41
publication: Kyoto RIMS Kôkyûroku
publication_status: published
publisher: Research Institute for Mathematical Sciences, Kyoto University
quality_controlled: '1'
status: public
title: Dirac masses and isometric rigidity
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 2125
year: '2019'
...