---
_id: '15025'
abstract:
- lang: eng
text: We consider quadratic forms of deterministic matrices A evaluated at the random
eigenvectors of a large N×N GOE or GUE matrix, or equivalently evaluated at the
columns of a Haar-orthogonal or Haar-unitary random matrix. We prove that, as
long as the deterministic matrix has rank much smaller than √N, the distributions
of the extrema of these quadratic forms are asymptotically the same as if the
eigenvectors were independent Gaussians. This reduces the problem to Gaussian
computations, which we carry out in several cases to illustrate our result, finding
Gumbel or Weibull limiting distributions depending on the signature of A. Our
result also naturally applies to the eigenvectors of any invariant ensemble.
acknowledgement: The first author was supported by the ERC Advanced Grant “RMTBeyond”
No. 101020331. The second author was supported by Fulbright Austria and the Austrian
Marshall Plan Foundation.
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: Benjamin
full_name: McKenna, Benjamin
id: b0cc634c-d549-11ee-96c8-87338c7ad808
last_name: McKenna
orcid: 0000-0003-2625-495X
citation:
ama: Erdös L, McKenna B. Extremal statistics of quadratic forms of GOE/GUE eigenvectors.
Annals of Applied Probability. 2024;34(1B):1623-1662. doi:10.1214/23-AAP2000
apa: Erdös, L., & McKenna, B. (2024). Extremal statistics of quadratic forms
of GOE/GUE eigenvectors. Annals of Applied Probability. Institute of Mathematical
Statistics. https://doi.org/10.1214/23-AAP2000
chicago: Erdös, László, and Benjamin McKenna. “Extremal Statistics of Quadratic
Forms of GOE/GUE Eigenvectors.” Annals of Applied Probability. Institute
of Mathematical Statistics, 2024. https://doi.org/10.1214/23-AAP2000.
ieee: L. Erdös and B. McKenna, “Extremal statistics of quadratic forms of GOE/GUE
eigenvectors,” Annals of Applied Probability, vol. 34, no. 1B. Institute
of Mathematical Statistics, pp. 1623–1662, 2024.
ista: Erdös L, McKenna B. 2024. Extremal statistics of quadratic forms of GOE/GUE
eigenvectors. Annals of Applied Probability. 34(1B), 1623–1662.
mla: Erdös, László, and Benjamin McKenna. “Extremal Statistics of Quadratic Forms
of GOE/GUE Eigenvectors.” Annals of Applied Probability, vol. 34, no. 1B,
Institute of Mathematical Statistics, 2024, pp. 1623–62, doi:10.1214/23-AAP2000.
short: L. Erdös, B. McKenna, Annals of Applied Probability 34 (2024) 1623–1662.
date_created: 2024-02-25T23:00:56Z
date_published: 2024-02-01T00:00:00Z
date_updated: 2024-02-27T08:29:05Z
day: '01'
department:
- _id: LaEr
doi: 10.1214/23-AAP2000
ec_funded: 1
external_id:
arxiv:
- '2208.12206'
intvolume: ' 34'
issue: 1B
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2208.12206
month: '02'
oa: 1
oa_version: Preprint
page: 1623-1662
project:
- _id: 62796744-2b32-11ec-9570-940b20777f1d
call_identifier: H2020
grant_number: '101020331'
name: Random matrices beyond Wigner-Dyson-Mehta
publication: 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: Extremal statistics of quadratic forms of GOE/GUE eigenvectors
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 34
year: '2024'
...
---
_id: '11741'
abstract:
- lang: eng
text: Following E. Wigner’s original vision, we prove that sampling the eigenvalue
gaps within the bulk spectrum of a fixed (deformed) Wigner matrix H yields the
celebrated Wigner-Dyson-Mehta universal statistics with high probability. Similarly,
we prove universality for a monoparametric family of deformed Wigner matrices
H+xA with a deterministic Hermitian matrix A and a fixed Wigner matrix H, just
using the randomness of a single scalar real random variable x. Both results constitute
quenched versions of bulk universality that has so far only been proven in annealed
sense with respect to the probability space of the matrix ensemble.
acknowledgement: "The authors are indebted to Sourav Chatterjee for forwarding the
very inspiring question that Stephen Shenker originally addressed to him which initiated
the current paper. They are also grateful that the authors of [23] kindly shared
their preliminary numerical results in June 2021.\r\nOpen access funding provided
by Institute of Science and Technology (IST Austria)."
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
citation:
ama: Cipolloni G, Erdös L, Schröder DJ. Quenched universality for deformed Wigner
matrices. Probability Theory and Related Fields. 2023;185:1183–1218. doi:10.1007/s00440-022-01156-7
apa: Cipolloni, G., Erdös, L., & Schröder, D. J. (2023). Quenched universality
for deformed Wigner matrices. Probability Theory and Related Fields. Springer
Nature. https://doi.org/10.1007/s00440-022-01156-7
chicago: Cipolloni, Giorgio, László Erdös, and Dominik J Schröder. “Quenched Universality
for Deformed Wigner Matrices.” Probability Theory and Related Fields. Springer
Nature, 2023. https://doi.org/10.1007/s00440-022-01156-7.
ieee: G. Cipolloni, L. Erdös, and D. J. Schröder, “Quenched universality for deformed
Wigner matrices,” Probability Theory and Related Fields, vol. 185. Springer
Nature, pp. 1183–1218, 2023.
ista: Cipolloni G, Erdös L, Schröder DJ. 2023. Quenched universality for deformed
Wigner matrices. Probability Theory and Related Fields. 185, 1183–1218.
mla: Cipolloni, Giorgio, et al. “Quenched Universality for Deformed Wigner Matrices.”
Probability Theory and Related Fields, vol. 185, Springer Nature, 2023,
pp. 1183–1218, doi:10.1007/s00440-022-01156-7.
short: G. Cipolloni, L. Erdös, D.J. Schröder, Probability Theory and Related Fields
185 (2023) 1183–1218.
date_created: 2022-08-07T22:02:00Z
date_published: 2023-04-01T00:00:00Z
date_updated: 2023-08-14T12:48:09Z
day: '01'
ddc:
- '510'
department:
- _id: LaEr
doi: 10.1007/s00440-022-01156-7
external_id:
arxiv:
- '2106.10200'
isi:
- '000830344500001'
file:
- access_level: open_access
checksum: b9247827dae5544d1d19c37abe547abc
content_type: application/pdf
creator: dernst
date_created: 2023-08-14T12:47:32Z
date_updated: 2023-08-14T12:47:32Z
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file_name: 2023_ProbabilityTheory_Cipolloni.pdf
file_size: 782278
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has_accepted_license: '1'
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language:
- iso: eng
month: '04'
oa: 1
oa_version: Published Version
page: 1183–1218
publication: Probability Theory and Related Fields
publication_identifier:
eissn:
- 1432-2064
issn:
- 0178-8051
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Quenched universality for deformed 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: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 185
year: '2023'
...
---
_id: '10405'
abstract:
- lang: eng
text: 'We consider large non-Hermitian random matrices X with complex, independent,
identically distributed centred entries and show that the linear statistics of
their eigenvalues are asymptotically Gaussian for test functions having 2+ϵ derivatives.
Previously this result was known only for a few special cases; either the test
functions were required to be analytic [72], or the distribution of the matrix
elements needed to be Gaussian [73], or at least match the Gaussian up to the
first four moments [82, 56]. We find the exact dependence of the limiting variance
on the fourth cumulant that was not known before. The proof relies on two novel
ingredients: (i) a local law for a product of two resolvents of the Hermitisation
of X with different spectral parameters and (ii) a coupling of several weakly
dependent Dyson Brownian motions. These methods are also the key inputs for our
analogous results on the linear eigenvalue statistics of real matrices X that
are presented in the companion paper [32]. '
acknowledgement: L.E. would like to thank Nathanaël Berestycki and D.S.would like
to thank Nina Holden for valuable discussions on the Gaussian freefield.G.C. and
L.E. are partially supported by ERC Advanced Grant No. 338804.G.C. received funding
from the European Union’s Horizon 2020 research and in-novation programme under
the Marie Skłodowska-Curie Grant Agreement No.665385. D.S. is supported by Dr. Max
Rössler, the Walter Haefner Foundation, and the ETH Zürich Foundation.
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
citation:
ama: Cipolloni G, Erdös L, Schröder DJ. Central limit theorem for linear eigenvalue
statistics of non-Hermitian random matrices. Communications on Pure and Applied
Mathematics. 2023;76(5):946-1034. doi:10.1002/cpa.22028
apa: Cipolloni, G., Erdös, L., & Schröder, D. J. (2023). Central limit theorem
for linear eigenvalue statistics of non-Hermitian random matrices. Communications
on Pure and Applied Mathematics. Wiley. https://doi.org/10.1002/cpa.22028
chicago: Cipolloni, Giorgio, László Erdös, and Dominik J Schröder. “Central Limit
Theorem for Linear Eigenvalue Statistics of Non-Hermitian Random Matrices.” Communications
on Pure and Applied Mathematics. Wiley, 2023. https://doi.org/10.1002/cpa.22028.
ieee: G. Cipolloni, L. Erdös, and D. J. Schröder, “Central limit theorem for linear
eigenvalue statistics of non-Hermitian random matrices,” Communications on
Pure and Applied Mathematics, vol. 76, no. 5. Wiley, pp. 946–1034, 2023.
ista: Cipolloni G, Erdös L, Schröder DJ. 2023. Central limit theorem for linear
eigenvalue statistics of non-Hermitian random matrices. Communications on Pure
and Applied Mathematics. 76(5), 946–1034.
mla: Cipolloni, Giorgio, et al. “Central Limit Theorem for Linear Eigenvalue Statistics
of Non-Hermitian Random Matrices.” Communications on Pure and Applied Mathematics,
vol. 76, no. 5, Wiley, 2023, pp. 946–1034, doi:10.1002/cpa.22028.
short: G. Cipolloni, L. Erdös, D.J. Schröder, Communications on Pure and Applied
Mathematics 76 (2023) 946–1034.
date_created: 2021-12-05T23:01:41Z
date_published: 2023-05-01T00:00:00Z
date_updated: 2023-10-04T09:22:55Z
day: '01'
ddc:
- '510'
department:
- _id: LaEr
doi: 10.1002/cpa.22028
ec_funded: 1
external_id:
arxiv:
- '1912.04100'
isi:
- '000724652500001'
file:
- access_level: open_access
checksum: 8346bc2642afb4ccb7f38979f41df5d9
content_type: application/pdf
creator: dernst
date_created: 2023-10-04T09:21:48Z
date_updated: 2023-10-04T09:21:48Z
file_id: '14388'
file_name: 2023_CommPureMathematics_Cipolloni.pdf
file_size: 803440
relation: main_file
success: 1
file_date_updated: 2023-10-04T09:21:48Z
has_accepted_license: '1'
intvolume: ' 76'
isi: 1
issue: '5'
language:
- iso: eng
month: '05'
oa: 1
oa_version: Published Version
page: 946-1034
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: Communications on Pure and Applied Mathematics
publication_identifier:
eissn:
- 1097-0312
issn:
- 0010-3640
publication_status: published
publisher: Wiley
quality_controlled: '1'
scopus_import: '1'
status: public
title: Central limit theorem for linear eigenvalue statistics of non-Hermitian random
matrices
tmp:
image: /images/cc_by_nc_nd.png
legal_code_url: https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
name: Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International
(CC BY-NC-ND 4.0)
short: CC BY-NC-ND (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 76
year: '2023'
...
---
_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: '12792'
abstract:
- lang: eng
text: In the physics literature the spectral form factor (SFF), the squared Fourier
transform of the empirical eigenvalue density, is the most common tool to test
universality for disordered quantum systems, yet previous mathematical results
have been restricted only to two exactly solvable models (Forrester in J Stat
Phys 183:33, 2021. https://doi.org/10.1007/s10955-021-02767-5, Commun Math Phys
387:215–235, 2021. https://doi.org/10.1007/s00220-021-04193-w). We rigorously
prove the physics prediction on SFF up to an intermediate time scale for a large
class of random matrices using a robust method, the multi-resolvent local laws.
Beyond Wigner matrices we also consider the monoparametric ensemble and prove
that universality of SFF can already be triggered by a single random parameter,
supplementing the recently proven Wigner–Dyson universality (Cipolloni et al.
in Probab Theory Relat Fields, 2021. https://doi.org/10.1007/s00440-022-01156-7)
to larger spectral scales. Remarkably, extensive numerics indicates that our formulas
correctly predict the SFF in the entire slope-dip-ramp regime, as customarily
called in physics.
acknowledgement: "We are grateful to the authors of [25] for sharing with us their
insights and preliminary numerical results. We are especially thankful to Stephen
Shenker for very valuable advice over several email communications. Helpful comments
on the manuscript from Peter Forrester and from the anonymous referees are also
acknowledged.\r\nOpen access funding provided by Institute of Science and Technology
(IST Austria).\r\nLászló Erdős: Partially supported by ERC Advanced Grant \"RMTBeyond\"
No. 101020331. Dominik Schröder: Supported by Dr. Max Rössler, the Walter Haefner
Foundation and the ETH Zürich Foundation."
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
citation:
ama: Cipolloni G, Erdös L, Schröder DJ. On the spectral form factor for random matrices.
Communications in Mathematical Physics. 2023;401:1665-1700. doi:10.1007/s00220-023-04692-y
apa: Cipolloni, G., Erdös, L., & Schröder, D. J. (2023). On the spectral form
factor for random matrices. Communications in Mathematical Physics. Springer
Nature. https://doi.org/10.1007/s00220-023-04692-y
chicago: Cipolloni, Giorgio, László Erdös, and Dominik J Schröder. “On the Spectral
Form Factor for Random Matrices.” Communications in Mathematical Physics.
Springer Nature, 2023. https://doi.org/10.1007/s00220-023-04692-y.
ieee: G. Cipolloni, L. Erdös, and D. J. Schröder, “On the spectral form factor for
random matrices,” Communications in Mathematical Physics, vol. 401. Springer
Nature, pp. 1665–1700, 2023.
ista: Cipolloni G, Erdös L, Schröder DJ. 2023. On the spectral form factor for random
matrices. Communications in Mathematical Physics. 401, 1665–1700.
mla: Cipolloni, Giorgio, et al. “On the Spectral Form Factor for Random Matrices.”
Communications in Mathematical Physics, vol. 401, Springer Nature, 2023,
pp. 1665–700, doi:10.1007/s00220-023-04692-y.
short: G. Cipolloni, L. Erdös, D.J. Schröder, Communications in Mathematical Physics
401 (2023) 1665–1700.
date_created: 2023-04-02T22:01:11Z
date_published: 2023-07-01T00:00:00Z
date_updated: 2023-10-04T12:10:31Z
day: '01'
ddc:
- '510'
department:
- _id: LaEr
doi: 10.1007/s00220-023-04692-y
ec_funded: 1
external_id:
isi:
- '000957343500001'
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checksum: 72057940f76654050ca84a221f21786c
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creator: dernst
date_created: 2023-10-04T12:09:18Z
date_updated: 2023-10-04T12:09:18Z
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file_name: 2023_CommMathPhysics_Cipolloni.pdf
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success: 1
file_date_updated: 2023-10-04T12:09:18Z
has_accepted_license: '1'
intvolume: ' 401'
isi: 1
language:
- iso: eng
month: '07'
oa: 1
oa_version: Published Version
page: 1665-1700
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: On the spectral form factor for 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: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
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...