---
_id: '8373'
abstract:
- lang: eng
text: It is well known that special Kubo-Ando operator means admit divergence center
interpretations, moreover, they are also mean squared error estimators for certain
metrics on positive definite operators. In this paper we give a divergence center
interpretation for every symmetric Kubo-Ando mean. This characterization of the
symmetric means naturally leads to a definition of weighted and multivariate versions
of a large class of symmetric Kubo-Ando means. We study elementary properties
of these weighted multivariate means, and note in particular that in the special
case of the geometric mean we recover the weighted A#H-mean introduced by Kim,
Lawson, and Lim.
acknowledgement: "The authors are grateful to Milán Mosonyi for fruitful discussions
on the topic, and to the anonymous referee for his/her comments and suggestions.\r\nJ.
Pitrik was supported by the Hungarian Academy of Sciences Lendület-Momentum Grant
for Quantum Information Theory, No. 96 141, and by Hungarian National Research,
Development and Innovation Office (NKFIH) via grants no. K119442, no. K124152, and
no. KH129601. D. Virosztek was supported by the ISTFELLOW program of the Institute
of Science and Technology Austria (project code IC1027FELL01), by the European Union's
Horizon 2020 research and innovation program under the Marie Sklodowska-Curie Grant
Agreement No. 846294, and partially supported by the Hungarian National Research,
Development and Innovation Office (NKFIH) via grants no. K124152, and no. KH129601."
article_processing_charge: No
article_type: original
author:
- first_name: József
full_name: Pitrik, József
last_name: Pitrik
- first_name: Daniel
full_name: Virosztek, Daniel
id: 48DB45DA-F248-11E8-B48F-1D18A9856A87
last_name: Virosztek
orcid: 0000-0003-1109-5511
citation:
ama: Pitrik J, Virosztek D. A divergence center interpretation of general symmetric
Kubo-Ando means, and related weighted multivariate operator means. *Linear Algebra
and its Applications*. 2021;609:203-217. doi:10.1016/j.laa.2020.09.007
apa: Pitrik, J., & Virosztek, D. (2021). A divergence center interpretation
of general symmetric Kubo-Ando means, and related weighted multivariate operator
means. *Linear Algebra and Its Applications*. Elsevier. https://doi.org/10.1016/j.laa.2020.09.007
chicago: Pitrik, József, and Daniel Virosztek. “A Divergence Center Interpretation
of General Symmetric Kubo-Ando Means, and Related Weighted Multivariate Operator
Means.” *Linear Algebra and Its Applications*. Elsevier, 2021. https://doi.org/10.1016/j.laa.2020.09.007.
ieee: J. Pitrik and D. Virosztek, “A divergence center interpretation of general
symmetric Kubo-Ando means, and related weighted multivariate operator means,”
*Linear Algebra and its Applications*, vol. 609. Elsevier, pp. 203–217, 2021.
ista: Pitrik J, Virosztek D. 2021. A divergence center interpretation of general
symmetric Kubo-Ando means, and related weighted multivariate operator means. Linear
Algebra and its Applications. 609, 203–217.
mla: Pitrik, József, and Daniel Virosztek. “A Divergence Center Interpretation of
General Symmetric Kubo-Ando Means, and Related Weighted Multivariate Operator
Means.” *Linear Algebra and Its Applications*, vol. 609, Elsevier, 2021,
pp. 203–17, doi:10.1016/j.laa.2020.09.007.
short: J. Pitrik, D. Virosztek, Linear Algebra and Its Applications 609 (2021) 203–217.
date_created: 2020-09-11T08:35:50Z
date_published: 2021-01-15T00:00:00Z
date_updated: 2021-01-11T16:28:42Z
day: '15'
department:
- _id: LaEr
doi: 10.1016/j.laa.2020.09.007
ec_funded: 1
external_id:
arxiv:
- '2002.11678'
intvolume: ' 609'
keyword:
- Kubo-Ando mean
- weighted multivariate mean
- barycenter
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/2002.11678
month: '01'
oa: 1
oa_version: Preprint
page: 203-217
project:
- _id: 26A455A6-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '846294'
name: Geometric study of Wasserstein spaces and free probability
- _id: 25681D80-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '291734'
name: International IST Postdoc Fellowship Programme
publication: Linear Algebra and its Applications
publication_identifier:
issn:
- 0024-3795
publication_status: published
publisher: Elsevier
quality_controlled: '1'
status: public
title: A divergence center interpretation of general symmetric Kubo-Ando means, and
related weighted multivariate operator means
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 609
year: '2021'
...
---
_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:
- IST Austria Thesis
article_processing_charge: No
author:
- first_name: Giorgio
full_name: Cipolloni, Giorgio
id: 42198EFA-F248-11E8-B48F-1D18A9856A87
last_name: Cipolloni
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*.
IST Austria. https://doi.org/10.15479/AT:ISTA:9022
chicago: Cipolloni, Giorgio. “Fluctuations in the Spectrum of Random Matrices.”
IST Austria, 2021. https://doi.org/10.15479/AT:ISTA:9022.
ieee: G. Cipolloni, “Fluctuations in the spectrum of random matrices,” IST Austria,
2021.
ista: Cipolloni G. 2021. Fluctuations in the spectrum of random matrices. IST Austria.
mla: Cipolloni, Giorgio. *Fluctuations in the Spectrum of Random Matrices*.
IST Austria, 2021, doi:10.15479/AT:ISTA:9022.
short: G. Cipolloni, Fluctuations in the Spectrum of Random Matrices, IST Austria,
2021.
date_created: 2021-01-21T18:16:54Z
date_published: 2021-01-25T00:00:00Z
date_updated: 2021-02-04T14:51:51Z
day: '25'
ddc:
- '510'
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
relation: main_file
success: 1
- access_level: closed
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:
eissn:
- 2663-337X
publication_status: published
publisher: IST 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: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2021'
...
---
_id: '9230'
abstract:
- lang: eng
text: "We consider a model of the Riemann zeta function on the critical axis and
study its maximum over intervals of length (log T)θ, where θ is either fixed or
tends to zero at a suitable rate.\r\nIt is shown that the deterministic level
of the maximum interpolates smoothly between the ones\r\nof log-correlated variables
and of i.i.d. random variables, exhibiting a smooth transition ‘from\r\n3/4 to
1/4’ in the second order. This provides a natural context where extreme value
statistics of\r\nlog-correlated variables with time-dependent variance and rate
occur. A key ingredient of the\r\nproof is a precise upper tail tightness estimate
for the maximum of the model on intervals of\r\nsize one, that includes a Gaussian
correction. This correction is expected to be present for the\r\nRiemann zeta
function and pertains to the question of the correct order of the maximum of\r\nthe
zeta function in large intervals."
acknowledgement: The research of L.-P. A. is supported in part by the grant NSF CAREER
DMS-1653602. G. D. gratefully acknowledges support from the European Union’s Horizon
2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement
No. 754411. The research of L. H. is supported in part by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) through Project-ID 233630050 -TRR 146, Project-ID
443891315 within SPP 2265 and Project-ID 446173099.
article_number: '2103.04817'
article_processing_charge: No
author:
- first_name: Louis-Pierre
full_name: Arguin, Louis-Pierre
last_name: Arguin
- first_name: Guillaume
full_name: Dubach, Guillaume
id: D5C6A458-10C4-11EA-ABF4-A4B43DDC885E
last_name: Dubach
orcid: 0000-0001-6892-8137
- first_name: Lisa
full_name: Hartung, Lisa
last_name: Hartung
citation:
ama: Arguin L-P, Dubach G, Hartung L. Maxima of a random model of the Riemann zeta
function over intervals of varying length. *arXiv*.
apa: Arguin, L.-P., Dubach, G., & Hartung, L. (n.d.). Maxima of a random model
of the Riemann zeta function over intervals of varying length. *arXiv*.
chicago: Arguin, Louis-Pierre, Guillaume Dubach, and Lisa Hartung. “Maxima of a
Random Model of the Riemann Zeta Function over Intervals of Varying Length.” *ArXiv*,
n.d.
ieee: L.-P. Arguin, G. Dubach, and L. Hartung, “Maxima of a random model of the
Riemann zeta function over intervals of varying length,” *arXiv*. .
ista: Arguin L-P, Dubach G, Hartung L. Maxima of a random model of the Riemann zeta
function over intervals of varying length. arXiv, 2103.04817.
mla: Arguin, Louis-Pierre, et al. “Maxima of a Random Model of the Riemann Zeta
Function over Intervals of Varying Length.” *ArXiv*, 2103.04817.
short: L.-P. Arguin, G. Dubach, L. Hartung, ArXiv (n.d.).
date_created: 2021-03-09T11:08:15Z
date_published: 2021-03-08T00:00:00Z
date_updated: 2021-03-09T12:07:57Z
day: '08'
department:
- _id: LaEr
ec_funded: 1
external_id:
arxiv:
- '2103.04817'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/2103.04817
month: '03'
oa: 1
oa_version: Preprint
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '754411'
name: ISTplus - Postdoctoral Fellowships
publication: arXiv
publication_status: submitted
status: public
title: Maxima of a random model of the Riemann zeta function over intervals of varying
length
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2021'
...
---
_id: '9281'
abstract:
- lang: eng
text: We comment on two formal proofs of Fermat's sum of two squares theorem, written
using the Mathematical Components libraries of the Coq proof assistant. The first
one follows Zagier's celebrated one-sentence proof; the second follows David Christopher's
recent new proof relying on partition-theoretic arguments. Both formal proofs
rely on a general property of involutions of finite sets, of independent interest.
The proof technique consists for the most part of automating recurrent tasks (such
as case distinctions and computations on natural numbers) via ad hoc tactics.
article_number: '2103.11389'
article_processing_charge: No
author:
- first_name: Guillaume
full_name: Dubach, Guillaume
id: D5C6A458-10C4-11EA-ABF4-A4B43DDC885E
last_name: Dubach
orcid: 0000-0001-6892-8137
- first_name: Fabian
full_name: Mühlböck, Fabian
id: 6395C5F6-89DF-11E9-9C97-6BDFE5697425
last_name: Mühlböck
orcid: 0000-0003-1548-0177
citation:
ama: Dubach G, Mühlböck F. Formal verification of Zagier’s one-sentence proof. *arXiv*.
apa: Dubach, G., & Mühlböck, F. (n.d.). Formal verification of Zagier’s one-sentence
proof. *arXiv*.
chicago: Dubach, Guillaume, and Fabian Mühlböck. “Formal Verification of Zagier’s
One-Sentence Proof.” *ArXiv*, n.d.
ieee: G. Dubach and F. Mühlböck, “Formal verification of Zagier’s one-sentence proof,”
*arXiv*. .
ista: Dubach G, Mühlböck F. Formal verification of Zagier’s one-sentence proof.
arXiv, 2103.11389.
mla: Dubach, Guillaume, and Fabian Mühlböck. “Formal Verification of Zagier’s One-Sentence
Proof.” *ArXiv*, 2103.11389.
short: G. Dubach, F. Mühlböck, ArXiv (n.d.).
date_created: 2021-03-23T05:38:48Z
date_published: 2021-03-21T00:00:00Z
date_updated: 2021-03-23T09:04:32Z
day: '21'
department:
- _id: LaEr
- _id: ToHe
ec_funded: 1
external_id:
arxiv:
- '2103.11389'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/2103.11389
month: '03'
oa: 1
oa_version: Preprint
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '754411'
name: ISTplus - Postdoctoral Fellowships
publication: arXiv
publication_status: draft
status: public
title: Formal verification of Zagier's one-sentence proof
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2021'
...
---
_id: '9036'
abstract:
- lang: eng
text: In this short note, we prove that the square root of the quantum Jensen-Shannon
divergence is a true metric on the cone of positive matrices, and hence in particular
on the quantum state space.
acknowledgement: D. Virosztek was supported by the European Union's Horizon 2020 research
and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 846294,
and partially supported by the Hungarian National Research, Development and Innovation
Office (NKFIH) via grants no. K124152, and no. KH129601.
article_number: '107595'
article_processing_charge: No
article_type: original
author:
- first_name: Daniel
full_name: Virosztek, Daniel
id: 48DB45DA-F248-11E8-B48F-1D18A9856A87
last_name: Virosztek
orcid: 0000-0003-1109-5511
citation:
ama: Virosztek D. The metric property of the quantum Jensen-Shannon divergence.
*Advances in Mathematics*. 2021;380(3). doi:10.1016/j.aim.2021.107595
apa: Virosztek, D. (2021). The metric property of the quantum Jensen-Shannon divergence.
*Advances in Mathematics*. Elsevier. https://doi.org/10.1016/j.aim.2021.107595
chicago: Virosztek, Daniel. “The Metric Property of the Quantum Jensen-Shannon Divergence.”
*Advances in Mathematics*. Elsevier, 2021. https://doi.org/10.1016/j.aim.2021.107595.
ieee: D. Virosztek, “The metric property of the quantum Jensen-Shannon divergence,”
*Advances in Mathematics*, vol. 380, no. 3. Elsevier, 2021.
ista: Virosztek D. 2021. The metric property of the quantum Jensen-Shannon divergence.
Advances in Mathematics. 380(3), 107595.
mla: Virosztek, Daniel. “The Metric Property of the Quantum Jensen-Shannon Divergence.”
*Advances in Mathematics*, vol. 380, no. 3, 107595, Elsevier, 2021, doi:10.1016/j.aim.2021.107595.
short: D. Virosztek, Advances in Mathematics 380 (2021).
date_created: 2021-01-22T17:55:17Z
date_published: 2021-03-26T00:00:00Z
date_updated: 2021-04-12T14:01:57Z
day: '26'
department:
- _id: LaEr
doi: 10.1016/j.aim.2021.107595
ec_funded: 1
external_id:
arxiv:
- '1910.10447'
intvolume: ' 380'
issue: '3'
keyword:
- General Mathematics
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1910.10447
month: '03'
oa: 1
oa_version: Preprint
project:
- _id: 26A455A6-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '846294'
name: Geometric study of Wasserstein spaces and free probability
publication: Advances in Mathematics
publication_identifier:
issn:
- 0001-8708
publication_status: published
publisher: Elsevier
quality_controlled: '1'
status: public
title: The metric property of the quantum Jensen-Shannon divergence
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 380
year: '2021'
...