---
_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:10.15479/AT:ISTA:9022
apa: Cipolloni, G. (2021). *Fluctuations in the spectrum of random matrices*.
Institute of Science and Technology Austria. https://doi.org/10.15479/AT:ISTA:9022
chicago: Cipolloni, Giorgio. “Fluctuations in the Spectrum of Random Matrices.”
Institute of Science and Technology Austria, 2021. https://doi.org/10.15479/AT:ISTA:9022.
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. *Fluctuations in the Spectrum of Random Matrices*.
Institute of Science and Technology Austria, 2021, doi:10.15479/AT:ISTA:9022.
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:
- access_level: open_access
checksum: 5a93658a5f19478372523ee232887e2b
content_type: application/pdf
creator: gcipollo
date_created: 2021-01-25T14:19:03Z
date_updated: 2021-01-25T14:19:03Z
file_id: '9043'
file_name: thesis.pdf
file_size: 4127796
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checksum: e8270eddfe6a988e92a53c88d1d19b8c
content_type: application/zip
creator: gcipollo
date_created: 2021-01-25T14:19:10Z
date_updated: 2021-01-25T14:19:10Z
file_id: '9044'
file_name: Thesis_files.zip
file_size: 12775206
relation: source_file
file_date_updated: 2021-01-25T14:19:10Z
has_accepted_license: '1'
language:
- iso: eng
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'
...