---
_id: '9022'
abstract:
- lang: eng
  text: "In the first part of the thesis we consider Hermitian random matrices. Firstly,
    we consider sample covariance matrices XX∗ with X having independent identically
    distributed (i.i.d.) centred entries. We prove a Central Limit Theorem for differences
    of linear statistics of XX∗ and its minor after removing the first column of X.
    Secondly, we consider Wigner-type matrices and prove that the eigenvalue statistics
    near cusp singularities of the limiting density of states are universal and that
    they form a Pearcey process. Since the limiting eigenvalue distribution admits
    only square root (edge) and cubic root (cusp) singularities, this concludes the
    third and last remaining case of the Wigner-Dyson-Mehta universality conjecture.
    The main technical ingredients are an optimal local law at the cusp, and the proof
    of the fast relaxation to equilibrium of the Dyson Brownian motion in the cusp
    regime.\r\nIn the second part we consider non-Hermitian matrices X with centred
    i.i.d. entries. We normalise the entries of X to have variance N −1. It is well
    known that the empirical eigenvalue density converges to the uniform distribution
    on the unit disk (circular law). In the first project, we prove universality of
    the local eigenvalue statistics close to the edge of the spectrum. This is the
    non-Hermitian analogue of the TracyWidom universality at the Hermitian edge. Technically
    we analyse the evolution of the spectral distribution of X along the Ornstein-Uhlenbeck
    flow for very long time\r\n(up to t = +∞). In the second project, we consider
    linear statistics of eigenvalues for macroscopic test functions f in the Sobolev
    space H2+ϵ and prove their convergence to the projection of the Gaussian Free
    Field on the unit disk. We prove this result for non-Hermitian matrices with real
    or complex entries. The main technical ingredients are: (i) local law for products
    of two resolvents at different spectral parameters, (ii) analysis of correlated
    Dyson Brownian motions.\r\nIn the third and final part we discuss the mathematically
    rigorous application of supersymmetric techniques (SUSY ) to give a lower tail
    estimate of the lowest singular value of X − z, with z ∈ C. More precisely, we
    use superbosonisation formula to give an integral representation of the resolvent
    of (X − z)(X − z)∗ which reduces to two and three contour integrals in the complex
    and real case, respectively. The rigorous analysis of these integrals is quite
    challenging since simple saddle point analysis cannot be applied (the main contribution
    comes from a non-trivial manifold). Our result\r\nimproves classical smoothing
    inequalities in the regime |z| ≈ 1; this result is essential to prove edge universality
    for i.i.d. non-Hermitian matrices."
acknowledgement: I gratefully acknowledge the financial support from the European
  Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie
  Grant Agreement No. 665385 and my advisor’s ERC Advanced Grant No. 338804.
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Giorgio
  full_name: Cipolloni, Giorgio
  id: 42198EFA-F248-11E8-B48F-1D18A9856A87
  last_name: Cipolloni
  orcid: 0000-0002-4901-7992
citation:
  ama: Cipolloni G. Fluctuations in the spectrum of random matrices. 2021. doi:<a
    href="https://doi.org/10.15479/AT:ISTA:9022">10.15479/AT:ISTA:9022</a>
  apa: Cipolloni, G. (2021). <i>Fluctuations in the spectrum of random matrices</i>.
    Institute of Science and Technology Austria. <a href="https://doi.org/10.15479/AT:ISTA:9022">https://doi.org/10.15479/AT:ISTA:9022</a>
  chicago: Cipolloni, Giorgio. “Fluctuations in the Spectrum of Random Matrices.”
    Institute of Science and Technology Austria, 2021. <a href="https://doi.org/10.15479/AT:ISTA:9022">https://doi.org/10.15479/AT:ISTA:9022</a>.
  ieee: G. Cipolloni, “Fluctuations in the spectrum of random matrices,” Institute
    of Science and Technology Austria, 2021.
  ista: Cipolloni G. 2021. Fluctuations in the spectrum of random matrices. Institute
    of Science and Technology Austria.
  mla: Cipolloni, Giorgio. <i>Fluctuations in the Spectrum of Random Matrices</i>.
    Institute of Science and Technology Austria, 2021, doi:<a href="https://doi.org/10.15479/AT:ISTA:9022">10.15479/AT:ISTA:9022</a>.
  short: G. Cipolloni, Fluctuations in the Spectrum of Random Matrices, Institute
    of Science and Technology Austria, 2021.
date_created: 2021-01-21T18:16:54Z
date_published: 2021-01-25T00:00:00Z
date_updated: 2023-09-07T13:29:32Z
day: '25'
ddc:
- '510'
degree_awarded: PhD
department:
- _id: GradSch
- _id: LaEr
doi: 10.15479/AT:ISTA:9022
ec_funded: 1
file:
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language:
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month: '01'
oa: 1
oa_version: Published Version
page: '380'
project:
- _id: 2564DBCA-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '665385'
  name: International IST Doctoral Program
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '338804'
  name: Random matrices, universality and disordered quantum systems
publication_identifier:
  issn:
  - 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria
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: Fluctuations in the spectrum of random matrices
type: dissertation
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
year: '2021'
...
---
_id: '6179'
abstract:
- lang: eng
  text: "In the first part of this thesis we consider large random matrices with arbitrary
    expectation and a general slowly decaying correlation among its entries. We prove
    universality of the local eigenvalue statistics and optimal local laws for the
    resolvent in the bulk and edge regime. The main novel tool is a systematic diagrammatic
    control of a multivariate cumulant expansion.\r\nIn the second part we consider
    Wigner-type matrices and show that at any cusp singularity of the limiting eigenvalue
    distribution the local eigenvalue statistics are uni- versal and form a Pearcey
    process. Since the density of states typically exhibits only square root or cubic
    root cusp singularities, our work complements previous results on the bulk and
    edge universality and it thus completes the resolution of the Wigner- Dyson-Mehta
    universality conjecture for the last remaining universality type. Our analysis
    holds not only for exact cusps, but approximate cusps as well, where an ex- tended
    Pearcey process emerges. As a main technical ingredient we prove an optimal local
    law at the cusp, and extend the fast relaxation to equilibrium of the Dyson Brow-
    nian motion to the cusp regime.\r\nIn the third and final part we explore the
    entrywise linear statistics of Wigner ma- trices and identify the fluctuations
    for a large class of test functions with little regularity. This enables us to
    study the rectangular Young diagram obtained from the interlacing eigenvalues
    of the random matrix and its minor, and we find that, despite having the same
    limit, the fluctuations differ from those of the algebraic Young tableaux equipped
    with the Plancharel measure."
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- 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: 'Schröder DJ. From Dyson to Pearcey: Universal statistics in random matrix
    theory. 2019. doi:<a href="https://doi.org/10.15479/AT:ISTA:th6179">10.15479/AT:ISTA:th6179</a>'
  apa: 'Schröder, D. J. (2019). <i>From Dyson to Pearcey: Universal statistics in
    random matrix theory</i>. Institute of Science and Technology Austria. <a href="https://doi.org/10.15479/AT:ISTA:th6179">https://doi.org/10.15479/AT:ISTA:th6179</a>'
  chicago: 'Schröder, Dominik J. “From Dyson to Pearcey: Universal Statistics in Random
    Matrix Theory.” Institute of Science and Technology Austria, 2019. <a href="https://doi.org/10.15479/AT:ISTA:th6179">https://doi.org/10.15479/AT:ISTA:th6179</a>.'
  ieee: 'D. J. Schröder, “From Dyson to Pearcey: Universal statistics in random matrix
    theory,” Institute of Science and Technology Austria, 2019.'
  ista: 'Schröder DJ. 2019. From Dyson to Pearcey: Universal statistics in random
    matrix theory. Institute of Science and Technology Austria.'
  mla: 'Schröder, Dominik J. <i>From Dyson to Pearcey: Universal Statistics in Random
    Matrix Theory</i>. Institute of Science and Technology Austria, 2019, doi:<a href="https://doi.org/10.15479/AT:ISTA:th6179">10.15479/AT:ISTA:th6179</a>.'
  short: 'D.J. Schröder, From Dyson to Pearcey: Universal Statistics in Random Matrix
    Theory, Institute of Science and Technology Austria, 2019.'
date_created: 2019-03-28T08:58:59Z
date_published: 2019-03-18T00:00:00Z
date_updated: 2024-02-22T14:34:33Z
day: '18'
ddc:
- '515'
- '519'
degree_awarded: PhD
department:
- _id: LaEr
doi: 10.15479/AT:ISTA:th6179
ec_funded: 1
file:
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  date_updated: 2020-07-14T12:47:21Z
  file_id: '6180'
  file_name: 2019_Schroeder_Thesis.tar.gz
  file_size: 7104482
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file_date_updated: 2020-07-14T12:47:21Z
has_accepted_license: '1'
language:
- iso: eng
month: '03'
oa: 1
oa_version: Published Version
page: '375'
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '338804'
  name: Random matrices, universality and disordered quantum systems
publication_identifier:
  issn:
  - 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria
related_material:
  record:
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    relation: part_of_dissertation
    status: public
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    relation: part_of_dissertation
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    relation: part_of_dissertation
    status: public
  - id: '6182'
    relation: part_of_dissertation
    status: public
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    relation: part_of_dissertation
    status: public
  - id: '6184'
    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: 'From Dyson to Pearcey: Universal statistics in random matrix theory'
type: dissertation
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
year: '2019'
...
---
_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:
- ISTA Thesis
article_processing_charge: No
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. 2018. doi:<a
    href="https://doi.org/10.15479/AT:ISTA:TH_1040">10.15479/AT:ISTA:TH_1040</a>
  apa: Alt, J. (2018). <i>Dyson equation and eigenvalue statistics of random matrices</i>.
    Institute of Science and Technology Austria. <a href="https://doi.org/10.15479/AT:ISTA:TH_1040">https://doi.org/10.15479/AT:ISTA:TH_1040</a>
  chicago: Alt, Johannes. “Dyson Equation and Eigenvalue Statistics of Random Matrices.”
    Institute of Science and Technology Austria, 2018. <a href="https://doi.org/10.15479/AT:ISTA:TH_1040">https://doi.org/10.15479/AT:ISTA:TH_1040</a>.
  ieee: J. Alt, “Dyson equation and eigenvalue statistics of random matrices,” Institute
    of Science and Technology Austria, 2018.
  ista: Alt J. 2018. Dyson equation and eigenvalue statistics of random matrices.
    Institute of Science and Technology Austria.
  mla: Alt, Johannes. <i>Dyson Equation and Eigenvalue Statistics of Random Matrices</i>.
    Institute of Science and Technology Austria, 2018, doi:<a href="https://doi.org/10.15479/AT:ISTA:TH_1040">10.15479/AT:ISTA:TH_1040</a>.
  short: J. Alt, Dyson Equation and Eigenvalue Statistics of Random Matrices, Institute
    of Science and Technology Austria, 2018.
date_created: 2018-12-11T11:44:53Z
date_published: 2018-07-12T00:00:00Z
date_updated: 2024-02-22T14:34:33Z
day: '12'
ddc:
- '515'
- '519'
degree_awarded: PhD
department:
- _id: LaEr
doi: 10.15479/AT:ISTA:TH_1040
ec_funded: 1
file:
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  date_updated: 2020-07-14T12:44:57Z
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file_date_updated: 2020-07-14T12:44:57Z
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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_identifier:
  issn:
  - 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria
publist_id: '7772'
pubrep_id: '1040'
related_material:
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    relation: part_of_dissertation
    status: public
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    relation: part_of_dissertation
    status: public
  - id: '566'
    relation: part_of_dissertation
    status: public
  - id: '1010'
    relation: part_of_dissertation
    status: public
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    relation: part_of_dissertation
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  - id: '6184'
    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: c635000d-4b10-11ee-a964-aac5a93f6ac1
year: '2018'
...