---
_id: '12707'
abstract:
- lang: eng
text: We establish precise right-tail small deviation estimates for the largest
eigenvalue of real symmetric and complex Hermitian matrices whose entries are
independent random variables with uniformly bounded moments. The proof relies
on a Green function comparison along a continuous interpolating matrix flow for
a long time. Less precise estimates are also obtained in the left tail.
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: Yuanyuan
full_name: Xu, Yuanyuan
id: 7902bdb1-a2a4-11eb-a164-c9216f71aea3
last_name: Xu
orcid: 0000-0003-1559-1205
citation:
ama: Erdös L, Xu Y. Small deviation estimates for the largest eigenvalue of Wigner
matrices. Bernoulli. 2023;29(2):1063-1079. doi:10.3150/22-BEJ1490
apa: Erdös, L., & Xu, Y. (2023). Small deviation estimates for the largest eigenvalue
of Wigner matrices. Bernoulli. Bernoulli Society for Mathematical Statistics
and Probability. https://doi.org/10.3150/22-BEJ1490
chicago: Erdös, László, and Yuanyuan Xu. “Small Deviation Estimates for the Largest
Eigenvalue of Wigner Matrices.” Bernoulli. Bernoulli Society for Mathematical
Statistics and Probability, 2023. https://doi.org/10.3150/22-BEJ1490.
ieee: L. Erdös and Y. Xu, “Small deviation estimates for the largest eigenvalue
of Wigner matrices,” Bernoulli, vol. 29, no. 2. Bernoulli Society for Mathematical
Statistics and Probability, pp. 1063–1079, 2023.
ista: Erdös L, Xu Y. 2023. Small deviation estimates for the largest eigenvalue
of Wigner matrices. Bernoulli. 29(2), 1063–1079.
mla: Erdös, László, and Yuanyuan Xu. “Small Deviation Estimates for the Largest
Eigenvalue of Wigner Matrices.” Bernoulli, vol. 29, no. 2, Bernoulli Society
for Mathematical Statistics and Probability, 2023, pp. 1063–79, doi:10.3150/22-BEJ1490.
short: L. Erdös, Y. Xu, Bernoulli 29 (2023) 1063–1079.
date_created: 2023-03-05T23:01:05Z
date_published: 2023-05-01T00:00:00Z
date_updated: 2023-10-04T10:21:07Z
day: '01'
department:
- _id: LaEr
doi: 10.3150/22-BEJ1490
ec_funded: 1
external_id:
arxiv:
- '2112.12093 '
isi:
- '000947270100008'
intvolume: ' 29'
isi: 1
issue: '2'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/2112.12093
month: '05'
oa: 1
oa_version: Preprint
page: 1063-1079
project:
- _id: 62796744-2b32-11ec-9570-940b20777f1d
call_identifier: H2020
grant_number: '101020331'
name: Random matrices beyond Wigner-Dyson-Mehta
publication: Bernoulli
publication_identifier:
issn:
- 1350-7265
publication_status: published
publisher: Bernoulli Society for Mathematical Statistics and Probability
quality_controlled: '1'
scopus_import: '1'
status: public
title: Small deviation estimates for the largest eigenvalue of Wigner matrices
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 29
year: '2023'
...
---
_id: '14775'
abstract:
- lang: eng
text: We establish a quantitative version of the Tracy–Widom law for the largest
eigenvalue of high-dimensional sample covariance matrices. To be precise, we show
that the fluctuations of the largest eigenvalue of a sample covariance matrix
X∗X converge to its Tracy–Widom limit at a rate nearly N−1/3, where X is an M×N
random matrix whose entries are independent real or complex random variables,
assuming that both M and N tend to infinity at a constant rate. This result improves
the previous estimate N−2/9 obtained by Wang (2019). Our proof relies on a Green
function comparison method (Adv. Math. 229 (2012) 1435–1515) using iterative cumulant
expansions, the local laws for the Green function and asymptotic properties of
the correlation kernel of the white Wishart ensemble.
acknowledgement: K. Schnelli was supported by the Swedish Research Council Grants
VR-2017-05195, and the Knut and Alice Wallenberg Foundation. Y. Xu was supported
by the Swedish Research Council Grant VR-2017-05195 and the ERC Advanced Grant “RMTBeyond”
No. 101020331.
article_processing_charge: No
article_type: original
author:
- first_name: Kevin
full_name: Schnelli, Kevin
id: 434AD0AE-F248-11E8-B48F-1D18A9856A87
last_name: Schnelli
orcid: 0000-0003-0954-3231
- first_name: Yuanyuan
full_name: Xu, Yuanyuan
id: 7902bdb1-a2a4-11eb-a164-c9216f71aea3
last_name: Xu
orcid: 0000-0003-1559-1205
citation:
ama: Schnelli K, Xu Y. Convergence rate to the Tracy–Widom laws for the largest
eigenvalue of sample covariance matrices. The Annals of Applied Probability.
2023;33(1):677-725. doi:10.1214/22-aap1826
apa: Schnelli, K., & Xu, Y. (2023). Convergence rate to the Tracy–Widom laws
for the largest eigenvalue of sample covariance matrices. The Annals of Applied
Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/22-aap1826
chicago: Schnelli, Kevin, and Yuanyuan Xu. “Convergence Rate to the Tracy–Widom
Laws for the Largest Eigenvalue of Sample Covariance Matrices.” The Annals
of Applied Probability. Institute of Mathematical Statistics, 2023. https://doi.org/10.1214/22-aap1826.
ieee: K. Schnelli and Y. Xu, “Convergence rate to the Tracy–Widom laws for the largest
eigenvalue of sample covariance matrices,” The Annals of Applied Probability,
vol. 33, no. 1. Institute of Mathematical Statistics, pp. 677–725, 2023.
ista: Schnelli K, Xu Y. 2023. Convergence rate to the Tracy–Widom laws for the largest
eigenvalue of sample covariance matrices. The Annals of Applied Probability. 33(1),
677–725.
mla: Schnelli, Kevin, and Yuanyuan Xu. “Convergence Rate to the Tracy–Widom Laws
for the Largest Eigenvalue of Sample Covariance Matrices.” The Annals of Applied
Probability, vol. 33, no. 1, Institute of Mathematical Statistics, 2023, pp.
677–725, doi:10.1214/22-aap1826.
short: K. Schnelli, Y. Xu, The Annals of Applied Probability 33 (2023) 677–725.
date_created: 2024-01-10T09:23:31Z
date_published: 2023-02-01T00:00:00Z
date_updated: 2024-01-10T13:31:46Z
day: '01'
department:
- _id: LaEr
doi: 10.1214/22-aap1826
ec_funded: 1
external_id:
arxiv:
- '2108.02728'
isi:
- '000946432400021'
intvolume: ' 33'
isi: 1
issue: '1'
keyword:
- Statistics
- Probability and Uncertainty
- Statistics and Probability
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2108.02728
month: '02'
oa: 1
oa_version: Preprint
page: 677-725
project:
- _id: 62796744-2b32-11ec-9570-940b20777f1d
call_identifier: H2020
grant_number: '101020331'
name: Random matrices beyond Wigner-Dyson-Mehta
publication: The Annals of Applied Probability
publication_identifier:
issn:
- 1050-5164
publication_status: published
publisher: Institute of Mathematical Statistics
quality_controlled: '1'
scopus_import: '1'
status: public
title: Convergence rate to the Tracy–Widom laws for the largest eigenvalue of sample
covariance matrices
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 33
year: '2023'
...
---
_id: '11332'
abstract:
- lang: eng
text: We show that the fluctuations of the largest eigenvalue of a real symmetric
or complex Hermitian Wigner matrix of size N converge to the Tracy–Widom laws
at a rate O(N^{-1/3+\omega }), as N tends to infinity. For Wigner matrices this
improves the previous rate O(N^{-2/9+\omega }) obtained by Bourgade (J Eur Math
Soc, 2021) for generalized Wigner matrices. Our result follows from a Green function
comparison theorem, originally introduced by Erdős et al. (Adv Math 229(3):1435–1515,
2012) to prove edge universality, on a finer spectral parameter scale with improved
error estimates. The proof relies on the continuous Green function flow induced
by a matrix-valued Ornstein–Uhlenbeck process. Precise estimates on leading contributions
from the third and fourth order moments of the matrix entries are obtained using
iterative cumulant expansions and recursive comparisons for correlation functions,
along with uniform convergence estimates for correlation kernels of the Gaussian
invariant ensembles.
acknowledgement: Kevin Schnelli is supported in parts by the Swedish Research Council
Grant VR-2017-05195, and the Knut and Alice Wallenberg Foundation. Yuanyuan Xu is
supported by the Swedish Research Council Grant VR-2017-05195 and the ERC Advanced
Grant “RMTBeyond” No. 101020331.
article_processing_charge: No
article_type: original
author:
- first_name: Kevin
full_name: Schnelli, Kevin
id: 434AD0AE-F248-11E8-B48F-1D18A9856A87
last_name: Schnelli
orcid: 0000-0003-0954-3231
- first_name: Yuanyuan
full_name: Xu, Yuanyuan
id: 7902bdb1-a2a4-11eb-a164-c9216f71aea3
last_name: Xu
citation:
ama: Schnelli K, Xu Y. Convergence rate to the Tracy–Widom laws for the largest
Eigenvalue of Wigner matrices. Communications in Mathematical Physics.
2022;393:839-907. doi:10.1007/s00220-022-04377-y
apa: Schnelli, K., & Xu, Y. (2022). Convergence rate to the Tracy–Widom laws
for the largest Eigenvalue of Wigner matrices. Communications in Mathematical
Physics. Springer Nature. https://doi.org/10.1007/s00220-022-04377-y
chicago: Schnelli, Kevin, and Yuanyuan Xu. “Convergence Rate to the Tracy–Widom
Laws for the Largest Eigenvalue of Wigner Matrices.” Communications in Mathematical
Physics. Springer Nature, 2022. https://doi.org/10.1007/s00220-022-04377-y.
ieee: K. Schnelli and Y. Xu, “Convergence rate to the Tracy–Widom laws for the largest
Eigenvalue of Wigner matrices,” Communications in Mathematical Physics,
vol. 393. Springer Nature, pp. 839–907, 2022.
ista: Schnelli K, Xu Y. 2022. Convergence rate to the Tracy–Widom laws for the largest
Eigenvalue of Wigner matrices. Communications in Mathematical Physics. 393, 839–907.
mla: Schnelli, Kevin, and Yuanyuan Xu. “Convergence Rate to the Tracy–Widom Laws
for the Largest Eigenvalue of Wigner Matrices.” Communications in Mathematical
Physics, vol. 393, Springer Nature, 2022, pp. 839–907, doi:10.1007/s00220-022-04377-y.
short: K. Schnelli, Y. Xu, Communications in Mathematical Physics 393 (2022) 839–907.
date_created: 2022-04-24T22:01:44Z
date_published: 2022-07-01T00:00:00Z
date_updated: 2023-08-03T06:34:24Z
day: '01'
ddc:
- '510'
department:
- _id: LaEr
doi: 10.1007/s00220-022-04377-y
ec_funded: 1
external_id:
arxiv:
- '2102.04330'
isi:
- '000782737200001'
file:
- access_level: open_access
checksum: bee0278c5efa9a33d9a2dc8d354a6c51
content_type: application/pdf
creator: dernst
date_created: 2022-08-05T06:01:13Z
date_updated: 2022-08-05T06:01:13Z
file_id: '11726'
file_name: 2022_CommunMathPhys_Schnelli.pdf
file_size: 1141462
relation: main_file
success: 1
file_date_updated: 2022-08-05T06:01:13Z
has_accepted_license: '1'
intvolume: ' 393'
isi: 1
language:
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license: https://creativecommons.org/licenses/by/4.0/
month: '07'
oa: 1
oa_version: Published Version
page: 839-907
project:
- _id: 62796744-2b32-11ec-9570-940b20777f1d
call_identifier: H2020
grant_number: '101020331'
name: Random matrices beyond Wigner-Dyson-Mehta
publication: Communications in Mathematical Physics
publication_identifier:
eissn:
- 1432-0916
issn:
- 0010-3616
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Convergence rate to the Tracy–Widom laws for the largest Eigenvalue of Wigner
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: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 393
year: '2022'
...
---
_id: '12243'
abstract:
- lang: eng
text: 'We consider the eigenvalues of a large dimensional real or complex Ginibre
matrix in the region of the complex plane where their real parts reach their maximum
value. This maximum follows the Gumbel distribution and that these extreme eigenvalues
form a Poisson point process as the dimension asymptotically tends to infinity.
In the complex case, these facts have already been established by Bender [Probab.
Theory Relat. Fields 147, 241 (2010)] and in the real case by Akemann and Phillips
[J. Stat. Phys. 155, 421 (2014)] even for the more general elliptic ensemble with
a sophisticated saddle point analysis. The purpose of this article is to give
a very short direct proof in the Ginibre case with an effective error term. Moreover,
our estimates on the correlation kernel in this regime serve as a key input for
accurately locating [Formula: see text] for any large matrix X with i.i.d. entries
in the companion paper [G. Cipolloni et al., arXiv:2206.04448 (2022)]. '
acknowledgement: "The authors are grateful to G. Akemann for bringing Refs. 19 and
24–26 to their attention. Discussions with Guillaume Dubach on a preliminary version
of this project are acknowledged.\r\nL.E. and Y.X. were supported by the ERC Advanced
Grant “RMTBeyond” under Grant No. 101020331. D.S. was supported by Dr. Max Rössler,
the Walter Haefner Foundation, and the ETH Zürich Foundation."
article_number: '103303'
article_processing_charge: Yes (via OA deal)
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
- first_name: Yuanyuan
full_name: Xu, Yuanyuan
id: 7902bdb1-a2a4-11eb-a164-c9216f71aea3
last_name: Xu
citation:
ama: Cipolloni G, Erdös L, Schröder DJ, Xu Y. Directional extremal statistics for
Ginibre eigenvalues. Journal of Mathematical Physics. 2022;63(10). doi:10.1063/5.0104290
apa: Cipolloni, G., Erdös, L., Schröder, D. J., & Xu, Y. (2022). Directional
extremal statistics for Ginibre eigenvalues. Journal of Mathematical Physics.
AIP Publishing. https://doi.org/10.1063/5.0104290
chicago: Cipolloni, Giorgio, László Erdös, Dominik J Schröder, and Yuanyuan Xu.
“Directional Extremal Statistics for Ginibre Eigenvalues.” Journal of Mathematical
Physics. AIP Publishing, 2022. https://doi.org/10.1063/5.0104290.
ieee: G. Cipolloni, L. Erdös, D. J. Schröder, and Y. Xu, “Directional extremal statistics
for Ginibre eigenvalues,” Journal of Mathematical Physics, vol. 63, no.
10. AIP Publishing, 2022.
ista: Cipolloni G, Erdös L, Schröder DJ, Xu Y. 2022. Directional extremal statistics
for Ginibre eigenvalues. Journal of Mathematical Physics. 63(10), 103303.
mla: Cipolloni, Giorgio, et al. “Directional Extremal Statistics for Ginibre Eigenvalues.”
Journal of Mathematical Physics, vol. 63, no. 10, 103303, AIP Publishing,
2022, doi:10.1063/5.0104290.
short: G. Cipolloni, L. Erdös, D.J. Schröder, Y. Xu, Journal of Mathematical Physics
63 (2022).
date_created: 2023-01-16T09:52:58Z
date_published: 2022-10-14T00:00:00Z
date_updated: 2023-08-04T09:40:02Z
day: '14'
ddc:
- '510'
- '530'
department:
- _id: LaEr
doi: 10.1063/5.0104290
ec_funded: 1
external_id:
arxiv:
- '2206.04443'
isi:
- '000869715800001'
file:
- access_level: open_access
checksum: 2db278ae5b07f345a7e3fec1f92b5c33
content_type: application/pdf
creator: dernst
date_created: 2023-01-30T08:01:10Z
date_updated: 2023-01-30T08:01:10Z
file_id: '12436'
file_name: 2022_JourMathPhysics_Cipolloni2.pdf
file_size: 7356807
relation: main_file
success: 1
file_date_updated: 2023-01-30T08:01:10Z
has_accepted_license: '1'
intvolume: ' 63'
isi: 1
issue: '10'
keyword:
- Mathematical Physics
- Statistical and Nonlinear Physics
language:
- iso: eng
month: '10'
oa: 1
oa_version: Published Version
project:
- _id: 62796744-2b32-11ec-9570-940b20777f1d
call_identifier: H2020
grant_number: '101020331'
name: Random matrices beyond Wigner-Dyson-Mehta
publication: Journal of Mathematical Physics
publication_identifier:
eissn:
- 1089-7658
issn:
- 0022-2488
publication_status: published
publisher: AIP Publishing
quality_controlled: '1'
scopus_import: '1'
status: public
title: Directional extremal statistics for Ginibre eigenvalues
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: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 63
year: '2022'
...