---
_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
arxiv: 1
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.
    <i>Annals of Applied Probability</i>. 2024;34(1B):1623-1662. doi:<a href="https://doi.org/10.1214/23-AAP2000">10.1214/23-AAP2000</a>
  apa: Erdös, L., &#38; McKenna, B. (2024). Extremal statistics of quadratic forms
    of GOE/GUE eigenvectors. <i>Annals of Applied Probability</i>. Institute of Mathematical
    Statistics. <a href="https://doi.org/10.1214/23-AAP2000">https://doi.org/10.1214/23-AAP2000</a>
  chicago: Erdös, László, and Benjamin McKenna. “Extremal Statistics of Quadratic
    Forms of GOE/GUE Eigenvectors.” <i>Annals of Applied Probability</i>. Institute
    of Mathematical Statistics, 2024. <a href="https://doi.org/10.1214/23-AAP2000">https://doi.org/10.1214/23-AAP2000</a>.
  ieee: L. Erdös and B. McKenna, “Extremal statistics of quadratic forms of GOE/GUE
    eigenvectors,” <i>Annals of Applied Probability</i>, 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.” <i>Annals of Applied Probability</i>, vol. 34, no. 1B,
    Institute of Mathematical Statistics, 2024, pp. 1623–62, doi:<a href="https://doi.org/10.1214/23-AAP2000">10.1214/23-AAP2000</a>.
  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: '14750'
abstract:
- lang: eng
  text: "Consider the random matrix model A1/2UBU∗A1/2, where A and B are two N ×
    N deterministic matrices and U is either an N × N Haar unitary or orthogonal random
    matrix. It is well known that on the macroscopic scale (Invent. Math. 104 (1991)
    201–220), the limiting empirical spectral distribution (ESD) of the above model
    is given by the free multiplicative convolution\r\nof the limiting ESDs of A and
    B, denoted as μα \x02 μβ, where μα and μβ are the limiting ESDs of A and B, respectively.
    In this paper, we study the asymptotic microscopic behavior of the edge eigenvalues
    and eigenvectors statistics. We prove that both the density of μA \x02μB, where
    μA and μB are the ESDs of A and B, respectively and the associated subordination
    functions\r\nhave a regular behavior near the edges. Moreover, we establish the
    local laws near the edges on the optimal scale. In particular, we prove that the
    entries of the resolvent are close to some functionals depending only on the eigenvalues
    of A, B and the subordination functions with optimal convergence rates. Our proofs
    and calculations are based on the techniques developed for the additive model
    A+UBU∗ in (J. Funct. Anal. 271 (2016) 672–719; Comm. Math.\r\nPhys. 349 (2017)
    947–990; Adv. Math. 319 (2017) 251–291; J. Funct. Anal. 279 (2020) 108639), and
    our results can be regarded as the counterparts of (J. Funct. Anal. 279 (2020)
    108639) for the multiplicative model. "
acknowledgement: "The first author is partially supported by NSF Grant DMS-2113489
  and grateful for the AMS-SIMONS travel grant (2020–2023). The second author is supported
  by the ERC Advanced Grant “RMTBeyond” No. 101020331.\r\nThe authors would like to
  thank the Editor, Associate Editor and an anonymous referee for their many critical
  suggestions which have significantly improved the paper. We also want to thank Zhigang
  Bao and Ji Oon Lee for many helpful discussions and comments."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Xiucai
  full_name: Ding, Xiucai
  last_name: Ding
- first_name: Hong Chang
  full_name: Ji, Hong Chang
  id: dd216c0a-c1f9-11eb-beaf-e9ea9d2de76d
  last_name: Ji
citation:
  ama: Ding X, Ji HC. Local laws for multiplication of random matrices. <i>The Annals
    of Applied Probability</i>. 2023;33(4):2981-3009. doi:<a href="https://doi.org/10.1214/22-aap1882">10.1214/22-aap1882</a>
  apa: Ding, X., &#38; Ji, H. C. (2023). Local laws for multiplication of random matrices.
    <i>The Annals of Applied Probability</i>. Institute of Mathematical Statistics.
    <a href="https://doi.org/10.1214/22-aap1882">https://doi.org/10.1214/22-aap1882</a>
  chicago: Ding, Xiucai, and Hong Chang Ji. “Local Laws for Multiplication of Random
    Matrices.” <i>The Annals of Applied Probability</i>. Institute of Mathematical
    Statistics, 2023. <a href="https://doi.org/10.1214/22-aap1882">https://doi.org/10.1214/22-aap1882</a>.
  ieee: X. Ding and H. C. Ji, “Local laws for multiplication of random matrices,”
    <i>The Annals of Applied Probability</i>, vol. 33, no. 4. Institute of Mathematical
    Statistics, pp. 2981–3009, 2023.
  ista: Ding X, Ji HC. 2023. Local laws for multiplication of random matrices. The
    Annals of Applied Probability. 33(4), 2981–3009.
  mla: Ding, Xiucai, and Hong Chang Ji. “Local Laws for Multiplication of Random Matrices.”
    <i>The Annals of Applied Probability</i>, vol. 33, no. 4, Institute of Mathematical
    Statistics, 2023, pp. 2981–3009, doi:<a href="https://doi.org/10.1214/22-aap1882">10.1214/22-aap1882</a>.
  short: X. Ding, H.C. Ji, The Annals of Applied Probability 33 (2023) 2981–3009.
date_created: 2024-01-08T13:03:18Z
date_published: 2023-08-01T00:00:00Z
date_updated: 2024-01-09T08:16:41Z
day: '01'
department:
- _id: LaEr
doi: 10.1214/22-aap1882
ec_funded: 1
external_id:
  arxiv:
  - '2010.16083'
intvolume: '        33'
issue: '4'
keyword:
- Statistics
- Probability and Uncertainty
- Statistics and Probability
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2010.16083
month: '08'
oa: 1
oa_version: Preprint
page: 2981-3009
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: Local laws for multiplication of random matrices
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 33
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
arxiv: 1
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. <i>The Annals of Applied Probability</i>.
    2023;33(1):677-725. doi:<a href="https://doi.org/10.1214/22-aap1826">10.1214/22-aap1826</a>
  apa: Schnelli, K., &#38; Xu, Y. (2023). Convergence rate to the Tracy–Widom laws
    for the largest eigenvalue of sample covariance matrices. <i>The Annals of Applied
    Probability</i>. Institute of Mathematical Statistics. <a href="https://doi.org/10.1214/22-aap1826">https://doi.org/10.1214/22-aap1826</a>
  chicago: Schnelli, Kevin, and Yuanyuan Xu. “Convergence Rate to the Tracy–Widom
    Laws for the Largest Eigenvalue of Sample Covariance Matrices.” <i>The Annals
    of Applied Probability</i>. Institute of Mathematical Statistics, 2023. <a href="https://doi.org/10.1214/22-aap1826">https://doi.org/10.1214/22-aap1826</a>.
  ieee: K. Schnelli and Y. Xu, “Convergence rate to the Tracy–Widom laws for the largest
    eigenvalue of sample covariance matrices,” <i>The Annals of Applied Probability</i>,
    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.” <i>The Annals of Applied
    Probability</i>, vol. 33, no. 1, Institute of Mathematical Statistics, 2023, pp.
    677–725, doi:<a href="https://doi.org/10.1214/22-aap1826">10.1214/22-aap1826</a>.
  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: '12761'
abstract:
- lang: eng
  text: "We consider the fluctuations of regular functions f of a Wigner matrix W
    viewed as an entire matrix f (W). Going beyond the well-studied tracial mode,
    Trf (W), which is equivalent to the customary linear statistics of eigenvalues,
    we show that Trf (W)A is asymptotically normal for any nontrivial bounded deterministic
    matrix A. We identify three different and asymptotically independent modes of
    this fluctuation, corresponding to the tracial part, the traceless diagonal part
    and the off-diagonal part of f (W) in the entire mesoscopic regime, where we find
    that the off-diagonal modes fluctuate on a much smaller scale than the tracial
    mode. As a main motivation to study CLT in such generality on small mesoscopic
    scales, we determine\r\nthe fluctuations in the eigenstate thermalization hypothesis
    (Phys. Rev. A 43 (1991) 2046–2049), that is, prove that the eigenfunction overlaps
    with any deterministic matrix are asymptotically Gaussian after a small spectral
    averaging. Finally, in the macroscopic regime our result also generalizes (Zh.
    Mat. Fiz. Anal. Geom. 9 (2013) 536–581, 611, 615) to complex W and to all crossover
    ensembles in between. The main technical inputs are the recent\r\nmultiresolvent
    local laws with traceless deterministic matrices from the companion paper (Comm.
    Math. Phys. 388 (2021) 1005–1048)."
acknowledgement: The second author is partially funded by the ERC Advanced Grant “RMTBEYOND”
  No. 101020331. The third author is supported by Dr. Max Rössler, the Walter Haefner
  Foundation and the ETH Zürich Foundation.
article_processing_charge: No
article_type: original
arxiv: 1
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. Functional central limit theorems for Wigner
    matrices. <i>Annals of Applied Probability</i>. 2023;33(1):447-489. doi:<a href="https://doi.org/10.1214/22-AAP1820">10.1214/22-AAP1820</a>
  apa: Cipolloni, G., Erdös, L., &#38; Schröder, D. J. (2023). Functional central
    limit theorems for Wigner matrices. <i>Annals of Applied Probability</i>. Institute
    of Mathematical Statistics. <a href="https://doi.org/10.1214/22-AAP1820">https://doi.org/10.1214/22-AAP1820</a>
  chicago: Cipolloni, Giorgio, László Erdös, and Dominik J Schröder. “Functional Central
    Limit Theorems for Wigner Matrices.” <i>Annals of Applied Probability</i>. Institute
    of Mathematical Statistics, 2023. <a href="https://doi.org/10.1214/22-AAP1820">https://doi.org/10.1214/22-AAP1820</a>.
  ieee: G. Cipolloni, L. Erdös, and D. J. Schröder, “Functional central limit theorems
    for Wigner matrices,” <i>Annals of Applied Probability</i>, vol. 33, no. 1. Institute
    of Mathematical Statistics, pp. 447–489, 2023.
  ista: Cipolloni G, Erdös L, Schröder DJ. 2023. Functional central limit theorems
    for Wigner matrices. Annals of Applied Probability. 33(1), 447–489.
  mla: Cipolloni, Giorgio, et al. “Functional Central Limit Theorems for Wigner Matrices.”
    <i>Annals of Applied Probability</i>, vol. 33, no. 1, Institute of Mathematical
    Statistics, 2023, pp. 447–89, doi:<a href="https://doi.org/10.1214/22-AAP1820">10.1214/22-AAP1820</a>.
  short: G. Cipolloni, L. Erdös, D.J. Schröder, Annals of Applied Probability 33 (2023)
    447–489.
date_created: 2023-03-26T22:01:08Z
date_published: 2023-02-01T00:00:00Z
date_updated: 2023-10-17T12:48:52Z
day: '01'
department:
- _id: LaEr
doi: 10.1214/22-AAP1820
ec_funded: 1
external_id:
  arxiv:
  - '2012.13218'
  isi:
  - '000946432400015'
intvolume: '        33'
isi: 1
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/2012.13218
month: '02'
oa: 1
oa_version: Preprint
page: 447-489
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: Functional central limit theorems for Wigner matrices
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 33
year: '2023'
...
---
_id: '10548'
abstract:
- lang: eng
  text: "Consider a linear elliptic partial differential equation in divergence form
    with a random coefficient field. The solution operator displays fluctuations around
    its expectation. The recently developed pathwise theory of fluctuations in stochastic
    homogenization reduces the characterization of these fluctuations to those of
    the so-called standard homogenization commutator. In this contribution, we investigate
    the scaling limit of this key quantity: starting\r\nfrom a Gaussian-like coefficient
    field with possibly strong correlations, we establish the convergence of the rescaled
    commutator to a fractional Gaussian field, depending on the decay of correlations
    of the coefficient field, and we\r\ninvestigate the (non)degeneracy of the limit.
    This extends to general dimension $d\\ge1$ previous results so far limited to
    dimension $d=1$, and to the continuum setting with strong correlations recent
    results in the discrete iid case."
acknowledgement: The authors thank Ivan Nourdin and Felix Otto for inspiring discussions.
  The work of MD is financially supported by the CNRS-Momentum program. Financial
  support of AG is acknowledged from the European Research Council under the European
  Community’s Seventh Framework Programme (FP7/2014-2019 Grant Agreement QUANTHOM
  335410).
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Mitia
  full_name: Duerinckx, Mitia
  last_name: Duerinckx
- first_name: Julian L
  full_name: Fischer, Julian L
  id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87
  last_name: Fischer
  orcid: 0000-0002-0479-558X
- first_name: Antoine
  full_name: Gloria, Antoine
  last_name: Gloria
citation:
  ama: Duerinckx M, Fischer JL, Gloria A. Scaling limit of the homogenization commutator
    for Gaussian coefficient  fields. <i>Annals of applied probability</i>. 2022;32(2):1179-1209.
    doi:<a href="https://doi.org/10.1214/21-AAP1705">10.1214/21-AAP1705</a>
  apa: Duerinckx, M., Fischer, J. L., &#38; Gloria, A. (2022). Scaling limit of the
    homogenization commutator for Gaussian coefficient  fields. <i>Annals of Applied
    Probability</i>. Institute of Mathematical Statistics. <a href="https://doi.org/10.1214/21-AAP1705">https://doi.org/10.1214/21-AAP1705</a>
  chicago: Duerinckx, Mitia, Julian L Fischer, and Antoine Gloria. “Scaling Limit
    of the Homogenization Commutator for Gaussian Coefficient  Fields.” <i>Annals
    of Applied Probability</i>. Institute of Mathematical Statistics, 2022. <a href="https://doi.org/10.1214/21-AAP1705">https://doi.org/10.1214/21-AAP1705</a>.
  ieee: M. Duerinckx, J. L. Fischer, and A. Gloria, “Scaling limit of the homogenization
    commutator for Gaussian coefficient  fields,” <i>Annals of applied probability</i>,
    vol. 32, no. 2. Institute of Mathematical Statistics, pp. 1179–1209, 2022.
  ista: Duerinckx M, Fischer JL, Gloria A. 2022. Scaling limit of the homogenization
    commutator for Gaussian coefficient  fields. Annals of applied probability. 32(2),
    1179–1209.
  mla: Duerinckx, Mitia, et al. “Scaling Limit of the Homogenization Commutator for
    Gaussian Coefficient  Fields.” <i>Annals of Applied Probability</i>, vol. 32,
    no. 2, Institute of Mathematical Statistics, 2022, pp. 1179–209, doi:<a href="https://doi.org/10.1214/21-AAP1705">10.1214/21-AAP1705</a>.
  short: M. Duerinckx, J.L. Fischer, A. Gloria, Annals of Applied Probability 32 (2022)
    1179–1209.
date_created: 2021-12-16T12:10:16Z
date_published: 2022-04-28T00:00:00Z
date_updated: 2023-08-02T13:35:06Z
day: '28'
department:
- _id: JuFi
doi: 10.1214/21-AAP1705
external_id:
  arxiv:
  - '1910.04088'
  isi:
  - '000791003700011'
intvolume: '        32'
isi: 1
issue: '2'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1910.04088
month: '04'
oa: 1
oa_version: Preprint
page: 1179-1209
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: Scaling limit of the homogenization commutator for Gaussian coefficient  fields
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 32
year: '2022'
...