---
_id: '1223'
abstract:
- lang: eng
text: We consider a random Schrödinger operator on the binary tree with a random
potential which is the sum of a random radially symmetric potential, Qr, and a
random transversally periodic potential, κQt, with coupling constant κ. Using
a new one-dimensional dynamical systems approach combined with Jensen's inequality
in hyperbolic space (our key estimate) we obtain a fractional moment estimate
proving localization for small and large κ. Together with a previous result we
therefore obtain a model with two Anderson transitions, from localization to delocalization
and back to localization, when increasing κ. As a by-product we also have a partially
new proof of one-dimensional Anderson localization at any disorder.
author:
- first_name: Richard
full_name: Froese, Richard
last_name: Froese
- first_name: Darrick
full_name: Lee, Darrick
last_name: Lee
- first_name: Christian
full_name: Sadel, Christian
id: 4760E9F8-F248-11E8-B48F-1D18A9856A87
last_name: Sadel
orcid: 0000-0001-8255-3968
- first_name: Wolfgang
full_name: Spitzer, Wolfgang
last_name: Spitzer
- first_name: Günter
full_name: Stolz, Günter
last_name: Stolz
citation:
ama: Froese R, Lee D, Sadel C, Spitzer W, Stolz G. Localization for transversally
periodic random potentials on binary trees. Journal of Spectral Theory.
2016;6(3):557-600. doi:10.4171/JST/132
apa: Froese, R., Lee, D., Sadel, C., Spitzer, W., & Stolz, G. (2016). Localization
for transversally periodic random potentials on binary trees. Journal of Spectral
Theory. European Mathematical Society. https://doi.org/10.4171/JST/132
chicago: Froese, Richard, Darrick Lee, Christian Sadel, Wolfgang Spitzer, and Günter
Stolz. “Localization for Transversally Periodic Random Potentials on Binary Trees.”
Journal of Spectral Theory. European Mathematical Society, 2016. https://doi.org/10.4171/JST/132.
ieee: R. Froese, D. Lee, C. Sadel, W. Spitzer, and G. Stolz, “Localization for transversally
periodic random potentials on binary trees,” Journal of Spectral Theory,
vol. 6, no. 3. European Mathematical Society, pp. 557–600, 2016.
ista: Froese R, Lee D, Sadel C, Spitzer W, Stolz G. 2016. Localization for transversally
periodic random potentials on binary trees. Journal of Spectral Theory. 6(3),
557–600.
mla: Froese, Richard, et al. “Localization for Transversally Periodic Random Potentials
on Binary Trees.” Journal of Spectral Theory, vol. 6, no. 3, European Mathematical
Society, 2016, pp. 557–600, doi:10.4171/JST/132.
short: R. Froese, D. Lee, C. Sadel, W. Spitzer, G. Stolz, Journal of Spectral Theory
6 (2016) 557–600.
date_created: 2018-12-11T11:50:48Z
date_published: 2016-01-01T00:00:00Z
date_updated: 2021-01-12T06:49:12Z
day: '01'
department:
- _id: LaEr
doi: 10.4171/JST/132
intvolume: ' 6'
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1408.3961
month: '01'
oa: 1
oa_version: Preprint
page: 557 - 600
publication: Journal of Spectral Theory
publication_status: published
publisher: European Mathematical Society
publist_id: '6112'
quality_controlled: '1'
scopus_import: 1
status: public
title: Localization for transversally periodic random potentials on binary trees
type: journal_article
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 6
year: '2016'
...
---
_id: '1257'
abstract:
- lang: eng
text: We consider products of random matrices that are small, independent identically
distributed perturbations of a fixed matrix (Formula presented.). Focusing on
the eigenvalues of (Formula presented.) of a particular size we obtain a limit
to a SDE in a critical scaling. Previous results required (Formula presented.)
to be a (conjugated) unitary matrix so it could not have eigenvalues of different
modulus. From the result we can also obtain a limit SDE for the Markov process
given by the action of the random products on the flag manifold. Applying the
result to random Schrödinger operators we can improve some results by Valko and
Virag showing GOE statistics for the rescaled eigenvalue process of a sequence
of Anderson models on long boxes. In particular, we solve a problem posed in their
work.
acknowledgement: Open access funding provided by Institute of Science and Technology
(IST Austria). The work of C. Sadel was supported by NSERC Discovery Grant 92997-2010
RGPIN and by the People Programme (Marie Curie Actions) of the EU 7th Framework
Programme FP7/2007-2013, REA Grant 291734.
article_processing_charge: Yes (via OA deal)
author:
- first_name: Christian
full_name: Sadel, Christian
id: 4760E9F8-F248-11E8-B48F-1D18A9856A87
last_name: Sadel
orcid: 0000-0001-8255-3968
- first_name: Bálint
full_name: Virág, Bálint
last_name: Virág
citation:
ama: Sadel C, Virág B. A central limit theorem for products of random matrices and
GOE statistics for the Anderson model on long boxes. Communications in Mathematical
Physics. 2016;343(3):881-919. doi:10.1007/s00220-016-2600-4
apa: Sadel, C., & Virág, B. (2016). A central limit theorem for products of
random matrices and GOE statistics for the Anderson model on long boxes. Communications
in Mathematical Physics. Springer. https://doi.org/10.1007/s00220-016-2600-4
chicago: Sadel, Christian, and Bálint Virág. “A Central Limit Theorem for Products
of Random Matrices and GOE Statistics for the Anderson Model on Long Boxes.” Communications
in Mathematical Physics. Springer, 2016. https://doi.org/10.1007/s00220-016-2600-4.
ieee: C. Sadel and B. Virág, “A central limit theorem for products of random matrices
and GOE statistics for the Anderson model on long boxes,” Communications in
Mathematical Physics, vol. 343, no. 3. Springer, pp. 881–919, 2016.
ista: Sadel C, Virág B. 2016. A central limit theorem for products of random matrices
and GOE statistics for the Anderson model on long boxes. Communications in Mathematical
Physics. 343(3), 881–919.
mla: Sadel, Christian, and Bálint Virág. “A Central Limit Theorem for Products of
Random Matrices and GOE Statistics for the Anderson Model on Long Boxes.” Communications
in Mathematical Physics, vol. 343, no. 3, Springer, 2016, pp. 881–919, doi:10.1007/s00220-016-2600-4.
short: C. Sadel, B. Virág, Communications in Mathematical Physics 343 (2016) 881–919.
date_created: 2018-12-11T11:50:59Z
date_published: 2016-05-01T00:00:00Z
date_updated: 2021-01-12T06:49:26Z
day: '01'
ddc:
- '510'
- '539'
department:
- _id: LaEr
doi: 10.1007/s00220-016-2600-4
ec_funded: 1
file:
- access_level: open_access
checksum: 4fb2411d9c2f56676123165aad46c828
content_type: application/pdf
creator: system
date_created: 2018-12-12T10:15:02Z
date_updated: 2020-07-14T12:44:42Z
file_id: '5119'
file_name: IST-2016-703-v1+1_s00220-016-2600-4.pdf
file_size: 800792
relation: main_file
file_date_updated: 2020-07-14T12:44:42Z
has_accepted_license: '1'
intvolume: ' 343'
issue: '3'
language:
- iso: eng
month: '05'
oa: 1
oa_version: Published Version
page: 881 - 919
project:
- _id: 25681D80-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '291734'
name: International IST Postdoc Fellowship Programme
- _id: B67AFEDC-15C9-11EA-A837-991A96BB2854
name: IST Austria Open Access Fund
publication: Communications in Mathematical Physics
publication_status: published
publisher: Springer
publist_id: '6067'
pubrep_id: '703'
quality_controlled: '1'
scopus_import: 1
status: public
title: A central limit theorem for products of random matrices and GOE statistics
for the Anderson model on long boxes
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: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 343
year: '2016'
...
---
_id: '1280'
abstract:
- lang: eng
text: We prove the Wigner-Dyson-Mehta conjecture at fixed energy in the bulk of
the spectrum for generalized symmetric and Hermitian Wigner matrices. Previous
results concerning the universality of random matrices either require an averaging
in the energy parameter or they hold only for Hermitian matrices if the energy
parameter is fixed. We develop a homogenization theory of the Dyson Brownian motion
and show that microscopic universality follows from mesoscopic statistics.
acknowledgement: "The work of P.B. was partially supported by National Sci-\r\nence
Foundation Grant DMS-1208859. The work of L.E. was partially supported\r\nby ERC
Advanced Grant RANMAT 338804. The work of H.-T. Y. was partially\r\nsupported by
National Science Foundation Grant DMS-1307444 and a Simons In-\r\nvestigator award.
\ The work of J.Y. was partially supported by National Science\r\nFoundation Grant
DMS-1207961. The major part of this research was conducted\r\nwhen all authors
were visiting IAS and were also supported by National Science\r\nFoundation Grant
DMS-1128255."
author:
- first_name: Paul
full_name: Bourgade, Paul
last_name: Bourgade
- first_name: László
full_name: Erdös, László
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
- first_name: Horngtzer
full_name: Yau, Horngtzer
last_name: Yau
- first_name: Jun
full_name: Yin, Jun
last_name: Yin
citation:
ama: Bourgade P, Erdös L, Yau H, Yin J. Fixed energy universality for generalized
wigner matrices. Communications on Pure and Applied Mathematics. 2016;69(10):1815-1881.
doi:10.1002/cpa.21624
apa: Bourgade, P., Erdös, L., Yau, H., & Yin, J. (2016). Fixed energy universality
for generalized wigner matrices. Communications on Pure and Applied Mathematics.
Wiley-Blackwell. https://doi.org/10.1002/cpa.21624
chicago: Bourgade, Paul, László Erdös, Horngtzer Yau, and Jun Yin. “Fixed Energy
Universality for Generalized Wigner Matrices.” Communications on Pure and Applied
Mathematics. Wiley-Blackwell, 2016. https://doi.org/10.1002/cpa.21624.
ieee: P. Bourgade, L. Erdös, H. Yau, and J. Yin, “Fixed energy universality for
generalized wigner matrices,” Communications on Pure and Applied Mathematics,
vol. 69, no. 10. Wiley-Blackwell, pp. 1815–1881, 2016.
ista: Bourgade P, Erdös L, Yau H, Yin J. 2016. Fixed energy universality for generalized
wigner matrices. Communications on Pure and Applied Mathematics. 69(10), 1815–1881.
mla: Bourgade, Paul, et al. “Fixed Energy Universality for Generalized Wigner Matrices.”
Communications on Pure and Applied Mathematics, vol. 69, no. 10, Wiley-Blackwell,
2016, pp. 1815–81, doi:10.1002/cpa.21624.
short: P. Bourgade, L. Erdös, H. Yau, J. Yin, Communications on Pure and Applied
Mathematics 69 (2016) 1815–1881.
date_created: 2018-12-11T11:51:07Z
date_published: 2016-10-01T00:00:00Z
date_updated: 2021-01-12T06:49:35Z
day: '01'
department:
- _id: LaEr
doi: 10.1002/cpa.21624
ec_funded: 1
intvolume: ' 69'
issue: '10'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1407.5606
month: '10'
oa: 1
oa_version: Preprint
page: 1815 - 1881
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '338804'
name: Random matrices, universality and disordered quantum systems
publication: Communications on Pure and Applied Mathematics
publication_status: published
publisher: Wiley-Blackwell
publist_id: '6036'
scopus_import: 1
status: public
title: Fixed energy universality for generalized wigner matrices
type: journal_article
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 69
year: '2016'
...
---
_id: '1674'
abstract:
- lang: eng
text: We consider N × N random matrices of the form H = W + V where W is a real
symmetric Wigner matrix and V a random or deterministic, real, diagonal matrix
whose entries are independent of W. We assume subexponential decay for the matrix
entries of W and we choose V so that the eigenvalues of W and V are typically
of the same order. For a large class of diagonal matrices V, we show that the
rescaled distribution of the extremal eigenvalues is given by the Tracy-Widom
distribution F1 in the limit of large N. Our proofs also apply to the complex
Hermitian setting, i.e. when W is a complex Hermitian Wigner matrix.
article_number: '1550018'
author:
- first_name: Jioon
full_name: Lee, Jioon
last_name: Lee
- first_name: Kevin
full_name: Schnelli, Kevin
id: 434AD0AE-F248-11E8-B48F-1D18A9856A87
last_name: Schnelli
orcid: 0000-0003-0954-3231
citation:
ama: Lee J, Schnelli K. Edge universality for deformed Wigner matrices. Reviews
in Mathematical Physics. 2015;27(8). doi:10.1142/S0129055X1550018X
apa: Lee, J., & Schnelli, K. (2015). Edge universality for deformed Wigner matrices.
Reviews in Mathematical Physics. World Scientific Publishing. https://doi.org/10.1142/S0129055X1550018X
chicago: Lee, Jioon, and Kevin Schnelli. “Edge Universality for Deformed Wigner
Matrices.” Reviews in Mathematical Physics. World Scientific Publishing,
2015. https://doi.org/10.1142/S0129055X1550018X.
ieee: J. Lee and K. Schnelli, “Edge universality for deformed Wigner matrices,”
Reviews in Mathematical Physics, vol. 27, no. 8. World Scientific Publishing,
2015.
ista: Lee J, Schnelli K. 2015. Edge universality for deformed Wigner matrices. Reviews
in Mathematical Physics. 27(8), 1550018.
mla: Lee, Jioon, and Kevin Schnelli. “Edge Universality for Deformed Wigner Matrices.”
Reviews in Mathematical Physics, vol. 27, no. 8, 1550018, World Scientific
Publishing, 2015, doi:10.1142/S0129055X1550018X.
short: J. Lee, K. Schnelli, Reviews in Mathematical Physics 27 (2015).
date_created: 2018-12-11T11:53:24Z
date_published: 2015-09-01T00:00:00Z
date_updated: 2021-01-12T06:52:26Z
day: '01'
department:
- _id: LaEr
doi: 10.1142/S0129055X1550018X
intvolume: ' 27'
issue: '8'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1407.8015
month: '09'
oa: 1
oa_version: Preprint
publication: Reviews in Mathematical Physics
publication_status: published
publisher: World Scientific Publishing
publist_id: '5475'
quality_controlled: '1'
scopus_import: 1
status: public
title: Edge universality for deformed Wigner matrices
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 27
year: '2015'
...
---
_id: '1677'
abstract:
- lang: eng
text: We consider real symmetric and complex Hermitian random matrices with the
additional symmetry hxy = hN-y,N-x. The matrix elements are independent (up to
the fourfold symmetry) and not necessarily identically distributed. This ensemble
naturally arises as the Fourier transform of a Gaussian orthogonal ensemble. Italso
occurs as the flip matrix model - an approximation of the two-dimensional Anderson
model at small disorder. We show that the density of states converges to the Wigner
semicircle law despite the new symmetry type. We also prove the local version
of the semicircle law on the optimal scale.
article_number: '103301'
author:
- first_name: Johannes
full_name: Alt, Johannes
id: 36D3D8B6-F248-11E8-B48F-1D18A9856A87
last_name: Alt
citation:
ama: Alt J. The local semicircle law for random matrices with a fourfold symmetry.
Journal of Mathematical Physics. 2015;56(10). doi:10.1063/1.4932606
apa: Alt, J. (2015). The local semicircle law for random matrices with a fourfold
symmetry. Journal of Mathematical Physics. American Institute of Physics.
https://doi.org/10.1063/1.4932606
chicago: Alt, Johannes. “The Local Semicircle Law for Random Matrices with a Fourfold
Symmetry.” Journal of Mathematical Physics. American Institute of Physics,
2015. https://doi.org/10.1063/1.4932606.
ieee: J. Alt, “The local semicircle law for random matrices with a fourfold symmetry,”
Journal of Mathematical Physics, vol. 56, no. 10. American Institute of
Physics, 2015.
ista: Alt J. 2015. The local semicircle law for random matrices with a fourfold
symmetry. Journal of Mathematical Physics. 56(10), 103301.
mla: Alt, Johannes. “The Local Semicircle Law for Random Matrices with a Fourfold
Symmetry.” Journal of Mathematical Physics, vol. 56, no. 10, 103301, American
Institute of Physics, 2015, doi:10.1063/1.4932606.
short: J. Alt, Journal of Mathematical Physics 56 (2015).
date_created: 2018-12-11T11:53:25Z
date_published: 2015-10-09T00:00:00Z
date_updated: 2023-09-07T12:38:08Z
day: '09'
department:
- _id: LaEr
doi: 10.1063/1.4932606
ec_funded: 1
intvolume: ' 56'
issue: '10'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1506.04683
month: '10'
oa: 1
oa_version: Preprint
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '338804'
name: Random matrices, universality and disordered quantum systems
publication: Journal of Mathematical Physics
publication_status: published
publisher: American Institute of Physics
publist_id: '5472'
quality_controlled: '1'
related_material:
record:
- id: '149'
relation: dissertation_contains
status: public
scopus_import: 1
status: public
title: The local semicircle law for random matrices with a fourfold symmetry
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 56
year: '2015'
...
---
_id: '1824'
abstract:
- lang: eng
text: Condensation phenomena arise through a collective behaviour of particles.
They are observed in both classical and quantum systems, ranging from the formation
of traffic jams in mass transport models to the macroscopic occupation of the
energetic ground state in ultra-cold bosonic gases (Bose-Einstein condensation).
Recently, it has been shown that a driven and dissipative system of bosons may
form multiple condensates. Which states become the condensates has, however, remained
elusive thus far. The dynamics of this condensation are described by coupled birth-death
processes, which also occur in evolutionary game theory. Here we apply concepts
from evolutionary game theory to explain the formation of multiple condensates
in such driven-dissipative bosonic systems. We show that the vanishing of relative
entropy production determines their selection. The condensation proceeds exponentially
fast, but the system never comes to rest. Instead, the occupation numbers of condensates
may oscillate, as we demonstrate for a rock-paper-scissors game of condensates.
article_number: '6977'
author:
- first_name: Johannes
full_name: Knebel, Johannes
last_name: Knebel
- first_name: Markus
full_name: Weber, Markus
last_name: Weber
- first_name: Torben H
full_name: Krüger, Torben H
id: 3020C786-F248-11E8-B48F-1D18A9856A87
last_name: Krüger
orcid: 0000-0002-4821-3297
- first_name: Erwin
full_name: Frey, Erwin
last_name: Frey
citation:
ama: Knebel J, Weber M, Krüger TH, Frey E. Evolutionary games of condensates in
coupled birth-death processes. Nature Communications. 2015;6. doi:10.1038/ncomms7977
apa: Knebel, J., Weber, M., Krüger, T. H., & Frey, E. (2015). Evolutionary games
of condensates in coupled birth-death processes. Nature Communications.
Nature Publishing Group. https://doi.org/10.1038/ncomms7977
chicago: Knebel, Johannes, Markus Weber, Torben H Krüger, and Erwin Frey. “Evolutionary
Games of Condensates in Coupled Birth-Death Processes.” Nature Communications.
Nature Publishing Group, 2015. https://doi.org/10.1038/ncomms7977.
ieee: J. Knebel, M. Weber, T. H. Krüger, and E. Frey, “Evolutionary games of condensates
in coupled birth-death processes,” Nature Communications, vol. 6. Nature
Publishing Group, 2015.
ista: Knebel J, Weber M, Krüger TH, Frey E. 2015. Evolutionary games of condensates
in coupled birth-death processes. Nature Communications. 6, 6977.
mla: Knebel, Johannes, et al. “Evolutionary Games of Condensates in Coupled Birth-Death
Processes.” Nature Communications, vol. 6, 6977, Nature Publishing Group,
2015, doi:10.1038/ncomms7977.
short: J. Knebel, M. Weber, T.H. Krüger, E. Frey, Nature Communications 6 (2015).
date_created: 2018-12-11T11:54:13Z
date_published: 2015-04-24T00:00:00Z
date_updated: 2021-01-12T06:53:26Z
day: '24'
ddc:
- '530'
department:
- _id: LaEr
doi: 10.1038/ncomms7977
file:
- access_level: open_access
checksum: c4cffb5c8b245e658a34eac71a03e7cc
content_type: application/pdf
creator: system
date_created: 2018-12-12T10:16:54Z
date_updated: 2020-07-14T12:45:17Z
file_id: '5245'
file_name: IST-2016-451-v1+1_ncomms7977.pdf
file_size: 1151501
relation: main_file
file_date_updated: 2020-07-14T12:45:17Z
has_accepted_license: '1'
intvolume: ' 6'
language:
- iso: eng
month: '04'
oa: 1
oa_version: Published Version
publication: Nature Communications
publication_status: published
publisher: Nature Publishing Group
publist_id: '5282'
pubrep_id: '451'
quality_controlled: '1'
scopus_import: 1
status: public
title: Evolutionary games of condensates in coupled birth-death processes
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: 6
year: '2015'
...
---
_id: '1864'
abstract:
- lang: eng
text: "The Altshuler–Shklovskii formulas (Altshuler and Shklovskii, BZh Eksp Teor
Fiz 91:200, 1986) predict, for any disordered quantum system in the diffusive
regime, a universal power law behaviour for the correlation functions of the mesoscopic
eigenvalue density. In this paper and its companion (Erdős and Knowles, The Altshuler–Shklovskii
formulas for random band matrices I: the unimodular case, 2013), we prove these
formulas for random band matrices. In (Erdős and Knowles, The Altshuler–Shklovskii
formulas for random band matrices I: the unimodular case, 2013) we introduced
a diagrammatic approach and presented robust estimates on general diagrams under
certain simplifying assumptions. In this paper, we remove these assumptions by
giving a general estimate of the subleading diagrams. We also give a precise analysis
of the leading diagrams which give rise to the Altschuler–Shklovskii power laws.
Moreover, we introduce a family of general random band matrices which interpolates
between real symmetric (β = 1) and complex Hermitian (β = 2) models, and track
the transition for the mesoscopic density–density correlation. Finally, we address
the higher-order correlation functions by proving that they behave asymptotically
according to a Gaussian process whose covariance is given by the Altshuler–Shklovskii
formulas.\r\n"
author:
- first_name: László
full_name: Erdös, László
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
- first_name: Antti
full_name: Knowles, Antti
last_name: Knowles
citation:
ama: 'Erdös L, Knowles A. The Altshuler–Shklovskii formulas for random band matrices
II: The general case. Annales Henri Poincare. 2015;16(3):709-799. doi:10.1007/s00023-014-0333-5'
apa: 'Erdös, L., & Knowles, A. (2015). The Altshuler–Shklovskii formulas for
random band matrices II: The general case. Annales Henri Poincare. Springer.
https://doi.org/10.1007/s00023-014-0333-5'
chicago: 'Erdös, László, and Antti Knowles. “The Altshuler–Shklovskii Formulas for
Random Band Matrices II: The General Case.” Annales Henri Poincare. Springer,
2015. https://doi.org/10.1007/s00023-014-0333-5.'
ieee: 'L. Erdös and A. Knowles, “The Altshuler–Shklovskii formulas for random band
matrices II: The general case,” Annales Henri Poincare, vol. 16, no. 3.
Springer, pp. 709–799, 2015.'
ista: 'Erdös L, Knowles A. 2015. The Altshuler–Shklovskii formulas for random band
matrices II: The general case. Annales Henri Poincare. 16(3), 709–799.'
mla: 'Erdös, László, and Antti Knowles. “The Altshuler–Shklovskii Formulas for Random
Band Matrices II: The General Case.” Annales Henri Poincare, vol. 16, no.
3, Springer, 2015, pp. 709–99, doi:10.1007/s00023-014-0333-5.'
short: L. Erdös, A. Knowles, Annales Henri Poincare 16 (2015) 709–799.
date_created: 2018-12-11T11:54:26Z
date_published: 2015-03-01T00:00:00Z
date_updated: 2021-01-12T06:53:42Z
day: '01'
department:
- _id: LaEr
doi: 10.1007/s00023-014-0333-5
ec_funded: 1
intvolume: ' 16'
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1309.5107
month: '03'
oa: 1
oa_version: Preprint
page: 709 - 799
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '338804'
name: Random matrices, universality and disordered quantum systems
publication: Annales Henri Poincare
publication_status: published
publisher: Springer
publist_id: '5233'
scopus_import: 1
status: public
title: 'The Altshuler–Shklovskii formulas for random band matrices II: The general
case'
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 16
year: '2015'
...
---
_id: '2166'
abstract:
- lang: eng
text: 'We consider the spectral statistics of large random band matrices on mesoscopic
energy scales. We show that the correlation function of the local eigenvalue density
exhibits a universal power law behaviour that differs from the Wigner-Dyson- Mehta
statistics. This law had been predicted in the physics literature by Altshuler
and Shklovskii in (Zh Eksp Teor Fiz (Sov Phys JETP) 91(64):220(127), 1986); it
describes the correlations of the eigenvalue density in general metallic sampleswith
weak disorder. Our result rigorously establishes the Altshuler-Shklovskii formulas
for band matrices. In two dimensions, where the leading term vanishes owing to
an algebraic cancellation, we identify the first non-vanishing term and show that
it differs substantially from the prediction of Kravtsov and Lerner in (Phys Rev
Lett 74:2563-2566, 1995). The proof is given in the current paper and its companion
(Ann. H. Poincaré. arXiv:1309.5107, 2014). '
author:
- first_name: László
full_name: Erdös, László
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
- first_name: Antti
full_name: Knowles, Antti
last_name: Knowles
citation:
ama: 'Erdös L, Knowles A. The Altshuler-Shklovskii formulas for random band matrices
I: the unimodular case. Communications in Mathematical Physics. 2015;333(3):1365-1416.
doi:10.1007/s00220-014-2119-5'
apa: 'Erdös, L., & Knowles, A. (2015). The Altshuler-Shklovskii formulas for
random band matrices I: the unimodular case. Communications in Mathematical
Physics. Springer. https://doi.org/10.1007/s00220-014-2119-5'
chicago: 'Erdös, László, and Antti Knowles. “The Altshuler-Shklovskii Formulas for
Random Band Matrices I: The Unimodular Case.” Communications in Mathematical
Physics. Springer, 2015. https://doi.org/10.1007/s00220-014-2119-5.'
ieee: 'L. Erdös and A. Knowles, “The Altshuler-Shklovskii formulas for random band
matrices I: the unimodular case,” Communications in Mathematical Physics,
vol. 333, no. 3. Springer, pp. 1365–1416, 2015.'
ista: 'Erdös L, Knowles A. 2015. The Altshuler-Shklovskii formulas for random band
matrices I: the unimodular case. Communications in Mathematical Physics. 333(3),
1365–1416.'
mla: 'Erdös, László, and Antti Knowles. “The Altshuler-Shklovskii Formulas for Random
Band Matrices I: The Unimodular Case.” Communications in Mathematical Physics,
vol. 333, no. 3, Springer, 2015, pp. 1365–416, doi:10.1007/s00220-014-2119-5.'
short: L. Erdös, A. Knowles, Communications in Mathematical Physics 333 (2015) 1365–1416.
date_created: 2018-12-11T11:56:05Z
date_published: 2015-02-01T00:00:00Z
date_updated: 2021-01-12T06:55:43Z
day: '01'
department:
- _id: LaEr
doi: 10.1007/s00220-014-2119-5
intvolume: ' 333'
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1309.5106
month: '02'
oa: 1
oa_version: Preprint
page: 1365 - 1416
publication: Communications in Mathematical Physics
publication_status: published
publisher: Springer
publist_id: '4818'
quality_controlled: '1'
scopus_import: 1
status: public
title: 'The Altshuler-Shklovskii formulas for random band matrices I: the unimodular
case'
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 333
year: '2015'
...
---
_id: '1505'
abstract:
- lang: eng
text: This paper is aimed at deriving the universality of the largest eigenvalue
of a class of high-dimensional real or complex sample covariance matrices of the
form W N =Σ 1/2XX∗Σ 1/2 . Here, X = (xij )M,N is an M× N random matrix with independent
entries xij , 1 ≤ i M,≤ 1 ≤ j ≤ N such that Exij = 0, E|xij |2 = 1/N . On dimensionality,
we assume that M = M(N) and N/M → d ε (0, ∞) as N ∞→. For a class of general deterministic
positive-definite M × M matrices Σ , under some additional assumptions on the
distribution of xij 's, we show that the limiting behavior of the largest eigenvalue
of W N is universal, via pursuing a Green function comparison strategy raised
in [Probab. Theory Related Fields 154 (2012) 341-407, Adv. Math. 229 (2012) 1435-1515]
by Erd″os, Yau and Yin for Wigner matrices and extended by Pillai and Yin [Ann.
Appl. Probab. 24 (2014) 935-1001] to sample covariance matrices in the null case
(&Epsi = I ). Consequently, in the standard complex case (Ex2 ij = 0), combing
this universality property and the results known for Gaussian matrices obtained
by El Karoui in [Ann. Probab. 35 (2007) 663-714] (nonsingular case) and Onatski
in [Ann. Appl. Probab. 18 (2008) 470-490] (singular case), we show that after
an appropriate normalization the largest eigenvalue of W N converges weakly to
the type 2 Tracy-Widom distribution TW2 . Moreover, in the real case, we show
that whenΣ is spiked with a fixed number of subcritical spikes, the type 1 Tracy-Widom
limit TW1 holds for the normalized largest eigenvalue of W N , which extends a
result of Féral and Péché in [J. Math. Phys. 50 (2009) 073302] to the scenario
of nondiagonal Σ and more generally distributed X . In summary, we establish the
Tracy-Widom type universality for the largest eigenvalue of generally distributed
sample covariance matrices under quite light assumptions on &Sigma . Applications
of these limiting results to statistical signal detection and structure recognition
of separable covariance matrices are also discussed.
acknowledgement: "B.Z. was supported in part by NSFC Grant 11071213, ZJNSF
\ Grant R6090034 and SRFDP Grant 20100101110001. P.G. was supported in part
by the Ministry of Education, Singapore, under Grant ARC 14/11. Z.W. was supported
\ in part by the Ministry of Education, Singapore, under Grant ARC 14/11,
\ and by a Grant R-155-000-131-112 at the National University of Singapore\r\n"
author:
- first_name: Zhigang
full_name: Bao, Zhigang
id: 442E6A6C-F248-11E8-B48F-1D18A9856A87
last_name: Bao
orcid: 0000-0003-3036-1475
- first_name: Guangming
full_name: Pan, Guangming
last_name: Pan
- first_name: Wang
full_name: Zhou, Wang
last_name: Zhou
citation:
ama: Bao Z, Pan G, Zhou W. Universality for the largest eigenvalue of sample covariance
matrices with general population. Annals of Statistics. 2015;43(1):382-421.
doi:10.1214/14-AOS1281
apa: Bao, Z., Pan, G., & Zhou, W. (2015). Universality for the largest eigenvalue
of sample covariance matrices with general population. Annals of Statistics.
Institute of Mathematical Statistics. https://doi.org/10.1214/14-AOS1281
chicago: Bao, Zhigang, Guangming Pan, and Wang Zhou. “Universality for the Largest
Eigenvalue of Sample Covariance Matrices with General Population.” Annals of
Statistics. Institute of Mathematical Statistics, 2015. https://doi.org/10.1214/14-AOS1281.
ieee: Z. Bao, G. Pan, and W. Zhou, “Universality for the largest eigenvalue of sample
covariance matrices with general population,” Annals of Statistics, vol.
43, no. 1. Institute of Mathematical Statistics, pp. 382–421, 2015.
ista: Bao Z, Pan G, Zhou W. 2015. Universality for the largest eigenvalue of sample
covariance matrices with general population. Annals of Statistics. 43(1), 382–421.
mla: Bao, Zhigang, et al. “Universality for the Largest Eigenvalue of Sample Covariance
Matrices with General Population.” Annals of Statistics, vol. 43, no. 1,
Institute of Mathematical Statistics, 2015, pp. 382–421, doi:10.1214/14-AOS1281.
short: Z. Bao, G. Pan, W. Zhou, Annals of Statistics 43 (2015) 382–421.
date_created: 2018-12-11T11:52:25Z
date_published: 2015-02-01T00:00:00Z
date_updated: 2021-01-12T06:51:14Z
day: '01'
department:
- _id: LaEr
doi: 10.1214/14-AOS1281
intvolume: ' 43'
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1304.5690
month: '02'
oa: 1
oa_version: Preprint
page: 382 - 421
publication: Annals of Statistics
publication_status: published
publisher: Institute of Mathematical Statistics
publist_id: '5672'
quality_controlled: '1'
status: public
title: Universality for the largest eigenvalue of sample covariance matrices with
general population
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 43
year: '2015'
...
---
_id: '1506'
abstract:
- lang: eng
text: Consider the square random matrix An = (aij)n,n, where {aij:= a(n)ij , i,
j = 1, . . . , n} is a collection of independent real random variables with means
zero and variances one. Under the additional moment condition supn max1≤i,j ≤n
Ea4ij <∞, we prove Girko's logarithmic law of det An in the sense that as n→∞
log | detAn| ? (1/2) log(n-1)! d/→√(1/2) log n N(0, 1).
author:
- first_name: Zhigang
full_name: Bao, Zhigang
id: 442E6A6C-F248-11E8-B48F-1D18A9856A87
last_name: Bao
orcid: 0000-0003-3036-1475
- first_name: Guangming
full_name: Pan, Guangming
last_name: Pan
- first_name: Wang
full_name: Zhou, Wang
last_name: Zhou
citation:
ama: Bao Z, Pan G, Zhou W. The logarithmic law of random determinant. Bernoulli.
2015;21(3):1600-1628. doi:10.3150/14-BEJ615
apa: Bao, Z., Pan, G., & Zhou, W. (2015). The logarithmic law of random determinant.
Bernoulli. Bernoulli Society for Mathematical Statistics and Probability.
https://doi.org/10.3150/14-BEJ615
chicago: Bao, Zhigang, Guangming Pan, and Wang Zhou. “The Logarithmic Law of Random
Determinant.” Bernoulli. Bernoulli Society for Mathematical Statistics
and Probability, 2015. https://doi.org/10.3150/14-BEJ615.
ieee: Z. Bao, G. Pan, and W. Zhou, “The logarithmic law of random determinant,”
Bernoulli, vol. 21, no. 3. Bernoulli Society for Mathematical Statistics
and Probability, pp. 1600–1628, 2015.
ista: Bao Z, Pan G, Zhou W. 2015. The logarithmic law of random determinant. Bernoulli.
21(3), 1600–1628.
mla: Bao, Zhigang, et al. “The Logarithmic Law of Random Determinant.” Bernoulli,
vol. 21, no. 3, Bernoulli Society for Mathematical Statistics and Probability,
2015, pp. 1600–28, doi:10.3150/14-BEJ615.
short: Z. Bao, G. Pan, W. Zhou, Bernoulli 21 (2015) 1600–1628.
date_created: 2018-12-11T11:52:25Z
date_published: 2015-08-01T00:00:00Z
date_updated: 2021-01-12T06:51:14Z
day: '01'
department:
- _id: LaEr
doi: 10.3150/14-BEJ615
intvolume: ' 21'
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1208.5823
month: '08'
oa: 1
oa_version: Preprint
page: 1600 - 1628
publication: Bernoulli
publication_status: published
publisher: Bernoulli Society for Mathematical Statistics and Probability
publist_id: '5671'
quality_controlled: '1'
status: public
title: The logarithmic law of random determinant
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 21
year: '2015'
...
---
_id: '1508'
abstract:
- lang: eng
text: We consider generalized Wigner ensembles and general β-ensembles with analytic
potentials for any β ≥ 1. The recent universality results in particular assert
that the local averages of consecutive eigenvalue gaps in the bulk of the spectrum
are universal in the sense that they coincide with those of the corresponding
Gaussian β-ensembles. In this article, we show that local averaging is not necessary
for this result, i.e. we prove that the single gap distributions in the bulk are
universal. In fact, with an additional step, our result can be extended to any
C4(ℝ) potential.
author:
- first_name: László
full_name: Erdös, László
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
- first_name: Horng
full_name: Yau, Horng
last_name: Yau
citation:
ama: Erdös L, Yau H. Gap universality of generalized Wigner and β ensembles. Journal
of the European Mathematical Society. 2015;17(8):1927-2036. doi:10.4171/JEMS/548
apa: Erdös, L., & Yau, H. (2015). Gap universality of generalized Wigner and
β ensembles. Journal of the European Mathematical Society. European Mathematical
Society. https://doi.org/10.4171/JEMS/548
chicago: Erdös, László, and Horng Yau. “Gap Universality of Generalized Wigner and
β Ensembles.” Journal of the European Mathematical Society. European Mathematical
Society, 2015. https://doi.org/10.4171/JEMS/548.
ieee: L. Erdös and H. Yau, “Gap universality of generalized Wigner and β ensembles,”
Journal of the European Mathematical Society, vol. 17, no. 8. European
Mathematical Society, pp. 1927–2036, 2015.
ista: Erdös L, Yau H. 2015. Gap universality of generalized Wigner and β ensembles.
Journal of the European Mathematical Society. 17(8), 1927–2036.
mla: Erdös, László, and Horng Yau. “Gap Universality of Generalized Wigner and β
Ensembles.” Journal of the European Mathematical Society, vol. 17, no.
8, European Mathematical Society, 2015, pp. 1927–2036, doi:10.4171/JEMS/548.
short: L. Erdös, H. Yau, Journal of the European Mathematical Society 17 (2015)
1927–2036.
date_created: 2018-12-11T11:52:26Z
date_published: 2015-08-01T00:00:00Z
date_updated: 2021-01-12T06:51:15Z
day: '01'
department:
- _id: LaEr
doi: 10.4171/JEMS/548
intvolume: ' 17'
issue: '8'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1211.3786
month: '08'
oa: 1
oa_version: Preprint
page: 1927 - 2036
publication: Journal of the European Mathematical Society
publication_status: published
publisher: European Mathematical Society
publist_id: '5669'
quality_controlled: '1'
scopus_import: 1
status: public
title: Gap universality of generalized Wigner and β ensembles
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 17
year: '2015'
...
---
_id: '1585'
abstract:
- lang: eng
text: In this paper, we consider the fluctuation of mutual information statistics
of a multiple input multiple output channel communication systems without assuming
that the entries of the channel matrix have zero pseudovariance. To this end,
we also establish a central limit theorem of the linear spectral statistics for
sample covariance matrices under general moment conditions by removing the restrictions
imposed on the second moment and fourth moment on the matrix entries in Bai and
Silverstein (2004).
acknowledgement: "G. Pan was supported by MOE Tier 2 under Grant 2014-T2-2-060 and
in part by Tier 1 under Grant RG25/14 through the Nanyang Technological University,
Singapore. W. Zhou was supported by the National University of Singapore, Singapore,
under Grant R-155-000-131-112.\r\n"
author:
- first_name: Zhigang
full_name: Bao, Zhigang
id: 442E6A6C-F248-11E8-B48F-1D18A9856A87
last_name: Bao
orcid: 0000-0003-3036-1475
- first_name: Guangming
full_name: Pan, Guangming
last_name: Pan
- first_name: Wang
full_name: Zhou, Wang
last_name: Zhou
citation:
ama: Bao Z, Pan G, Zhou W. Asymptotic mutual information statistics of MIMO channels
and CLT of sample covariance matrices. IEEE Transactions on Information Theory.
2015;61(6):3413-3426. doi:10.1109/TIT.2015.2421894
apa: Bao, Z., Pan, G., & Zhou, W. (2015). Asymptotic mutual information statistics
of MIMO channels and CLT of sample covariance matrices. IEEE Transactions on
Information Theory. IEEE. https://doi.org/10.1109/TIT.2015.2421894
chicago: Bao, Zhigang, Guangming Pan, and Wang Zhou. “Asymptotic Mutual Information
Statistics of MIMO Channels and CLT of Sample Covariance Matrices.” IEEE Transactions
on Information Theory. IEEE, 2015. https://doi.org/10.1109/TIT.2015.2421894.
ieee: Z. Bao, G. Pan, and W. Zhou, “Asymptotic mutual information statistics of
MIMO channels and CLT of sample covariance matrices,” IEEE Transactions on
Information Theory, vol. 61, no. 6. IEEE, pp. 3413–3426, 2015.
ista: Bao Z, Pan G, Zhou W. 2015. Asymptotic mutual information statistics of MIMO
channels and CLT of sample covariance matrices. IEEE Transactions on Information
Theory. 61(6), 3413–3426.
mla: Bao, Zhigang, et al. “Asymptotic Mutual Information Statistics of MIMO Channels
and CLT of Sample Covariance Matrices.” IEEE Transactions on Information Theory,
vol. 61, no. 6, IEEE, 2015, pp. 3413–26, doi:10.1109/TIT.2015.2421894.
short: Z. Bao, G. Pan, W. Zhou, IEEE Transactions on Information Theory 61 (2015)
3413–3426.
date_created: 2018-12-11T11:52:52Z
date_published: 2015-06-01T00:00:00Z
date_updated: 2021-01-12T06:51:46Z
day: '01'
department:
- _id: LaEr
doi: 10.1109/TIT.2015.2421894
intvolume: ' 61'
issue: '6'
language:
- iso: eng
month: '06'
oa_version: None
page: 3413 - 3426
publication: IEEE Transactions on Information Theory
publication_status: published
publisher: IEEE
publist_id: '5586'
quality_controlled: '1'
scopus_import: 1
status: public
title: Asymptotic mutual information statistics of MIMO channels and CLT of sample
covariance matrices
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 61
year: '2015'
...
---
_id: '2699'
abstract:
- lang: eng
text: "We prove the universality of the β-ensembles with convex analytic potentials
and for any β >\r\n0, i.e. we show that the spacing distributions of log-gases
at any inverse temperature β coincide with those of the Gaussian β-ensembles."
author:
- first_name: László
full_name: Erdös, László
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
- first_name: Paul
full_name: Bourgade, Paul
last_name: Bourgade
- first_name: Horng
full_name: Yau, Horng
last_name: Yau
citation:
ama: Erdös L, Bourgade P, Yau H. Universality of general β-ensembles. Duke Mathematical
Journal. 2014;163(6):1127-1190. doi:10.1215/00127094-2649752
apa: Erdös, L., Bourgade, P., & Yau, H. (2014). Universality of general β-ensembles.
Duke Mathematical Journal. Duke University Press. https://doi.org/10.1215/00127094-2649752
chicago: Erdös, László, Paul Bourgade, and Horng Yau. “Universality of General β-Ensembles.”
Duke Mathematical Journal. Duke University Press, 2014. https://doi.org/10.1215/00127094-2649752.
ieee: L. Erdös, P. Bourgade, and H. Yau, “Universality of general β-ensembles,”
Duke Mathematical Journal, vol. 163, no. 6. Duke University Press, pp.
1127–1190, 2014.
ista: Erdös L, Bourgade P, Yau H. 2014. Universality of general β-ensembles. Duke
Mathematical Journal. 163(6), 1127–1190.
mla: Erdös, László, et al. “Universality of General β-Ensembles.” Duke Mathematical
Journal, vol. 163, no. 6, Duke University Press, 2014, pp. 1127–90, doi:10.1215/00127094-2649752.
short: L. Erdös, P. Bourgade, H. Yau, Duke Mathematical Journal 163 (2014) 1127–1190.
date_created: 2018-12-11T11:59:08Z
date_published: 2014-04-01T00:00:00Z
date_updated: 2021-01-12T06:59:07Z
day: '01'
department:
- _id: LaEr
doi: 10.1215/00127094-2649752
intvolume: ' 163'
issue: '6'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1104.2272
month: '04'
oa: 1
oa_version: Preprint
page: 1127 - 1190
publication: Duke Mathematical Journal
publication_status: published
publisher: Duke University Press
publist_id: '4197'
quality_controlled: '1'
scopus_import: 1
status: public
title: Universality of general β-ensembles
type: journal_article
user_id: 3FFCCD3A-F248-11E8-B48F-1D18A9856A87
volume: 163
year: '2014'
...
---
_id: '1926'
abstract:
- lang: eng
text: We consider cross products of finite graphs with a class of trees that have
arbitrarily but finitely long line segments, such as the Fibonacci tree. Such
cross products are called tree-strips. We prove that for small disorder random
Schrödinger operators on such tree-strips have purely absolutely continuous spectrum
in a certain set.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Christian
full_name: Sadel, Christian
id: 4760E9F8-F248-11E8-B48F-1D18A9856A87
last_name: Sadel
orcid: 0000-0001-8255-3968
citation:
ama: Sadel C. Absolutely continuous spectrum for random Schrödinger operators on
the Fibonacci and similar Tree-strips. Mathematical Physics, Analysis and Geometry.
2014;17(3-4):409-440. doi:10.1007/s11040-014-9163-4
apa: Sadel, C. (2014). Absolutely continuous spectrum for random Schrödinger operators
on the Fibonacci and similar Tree-strips. Mathematical Physics, Analysis and
Geometry. Springer. https://doi.org/10.1007/s11040-014-9163-4
chicago: Sadel, Christian. “Absolutely Continuous Spectrum for Random Schrödinger
Operators on the Fibonacci and Similar Tree-Strips.” Mathematical Physics,
Analysis and Geometry. Springer, 2014. https://doi.org/10.1007/s11040-014-9163-4.
ieee: C. Sadel, “Absolutely continuous spectrum for random Schrödinger operators
on the Fibonacci and similar Tree-strips,” Mathematical Physics, Analysis and
Geometry, vol. 17, no. 3–4. Springer, pp. 409–440, 2014.
ista: Sadel C. 2014. Absolutely continuous spectrum for random Schrödinger operators
on the Fibonacci and similar Tree-strips. Mathematical Physics, Analysis and Geometry.
17(3–4), 409–440.
mla: Sadel, Christian. “Absolutely Continuous Spectrum for Random Schrödinger Operators
on the Fibonacci and Similar Tree-Strips.” Mathematical Physics, Analysis and
Geometry, vol. 17, no. 3–4, Springer, 2014, pp. 409–40, doi:10.1007/s11040-014-9163-4.
short: C. Sadel, Mathematical Physics, Analysis and Geometry 17 (2014) 409–440.
date_created: 2018-12-11T11:54:45Z
date_published: 2014-12-17T00:00:00Z
date_updated: 2021-01-12T06:54:07Z
day: '17'
department:
- _id: LaEr
doi: 10.1007/s11040-014-9163-4
ec_funded: 1
external_id:
arxiv:
- '1304.3862'
intvolume: ' 17'
issue: 3-4
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1304.3862
month: '12'
oa: 1
oa_version: Preprint
page: 409 - 440
project:
- _id: 26450934-B435-11E9-9278-68D0E5697425
name: NSERC Postdoctoral fellowship
- _id: 25681D80-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '291734'
name: International IST Postdoc Fellowship Programme
publication: Mathematical Physics, Analysis and Geometry
publication_status: published
publisher: Springer
publist_id: '5168'
quality_controlled: '1'
scopus_import: 1
status: public
title: Absolutely continuous spectrum for random Schrödinger operators on the Fibonacci
and similar Tree-strips
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 17
year: '2014'
...
---
_id: '1937'
abstract:
- lang: eng
text: We prove the edge universality of the beta ensembles for any β ≥ 1, provided
that the limiting spectrum is supported on a single interval, and the external
potential is C4 and regular. We also prove that the edge universality holds for
generalized Wigner matrices for all symmetry classes. Moreover, our results allow
us to extend bulk universality for beta ensembles from analytic potentials to
potentials in class C4.
author:
- first_name: Paul
full_name: Bourgade, Paul
last_name: Bourgade
- first_name: László
full_name: Erdös, László
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
- first_name: Horngtzer
full_name: Yau, Horngtzer
last_name: Yau
citation:
ama: Bourgade P, Erdös L, Yau H. Edge universality of beta ensembles. Communications
in Mathematical Physics. 2014;332(1):261-353. doi:10.1007/s00220-014-2120-z
apa: Bourgade, P., Erdös, L., & Yau, H. (2014). Edge universality of beta ensembles.
Communications in Mathematical Physics. Springer. https://doi.org/10.1007/s00220-014-2120-z
chicago: Bourgade, Paul, László Erdös, and Horngtzer Yau. “Edge Universality of
Beta Ensembles.” Communications in Mathematical Physics. Springer, 2014.
https://doi.org/10.1007/s00220-014-2120-z.
ieee: P. Bourgade, L. Erdös, and H. Yau, “Edge universality of beta ensembles,”
Communications in Mathematical Physics, vol. 332, no. 1. Springer, pp.
261–353, 2014.
ista: Bourgade P, Erdös L, Yau H. 2014. Edge universality of beta ensembles. Communications
in Mathematical Physics. 332(1), 261–353.
mla: Bourgade, Paul, et al. “Edge Universality of Beta Ensembles.” Communications
in Mathematical Physics, vol. 332, no. 1, Springer, 2014, pp. 261–353, doi:10.1007/s00220-014-2120-z.
short: P. Bourgade, L. Erdös, H. Yau, Communications in Mathematical Physics 332
(2014) 261–353.
date_created: 2018-12-11T11:54:48Z
date_published: 2014-11-01T00:00:00Z
date_updated: 2021-01-12T06:54:12Z
day: '01'
department:
- _id: LaEr
doi: 10.1007/s00220-014-2120-z
intvolume: ' 332'
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1306.5728
month: '11'
oa: 1
oa_version: Submitted Version
page: 261 - 353
project:
- _id: 25BDE9A4-B435-11E9-9278-68D0E5697425
grant_number: SFB-TR3-TP10B
name: Glutamaterge synaptische Übertragung und Plastizität in hippocampalen Mikroschaltkreisen
publication: Communications in Mathematical Physics
publication_status: published
publisher: Springer
publist_id: '5158'
quality_controlled: '1'
scopus_import: 1
status: public
title: Edge universality of beta ensembles
type: journal_article
user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87
volume: 332
year: '2014'
...
---
_id: '2019'
abstract:
- lang: eng
text: We prove that the empirical density of states of quantum spin glasses on arbitrary
graphs converges to a normal distribution as long as the maximal degree is negligible
compared with the total number of edges. This extends the recent results of Keating
et al. (2014) that were proved for graphs with bounded chromatic number and with
symmetric coupling distribution. Furthermore, we generalise the result to arbitrary
hypergraphs. We test the optimality of our condition on the maximal degree for
p-uniform hypergraphs that correspond to p-spin glass Hamiltonians acting on n
distinguishable spin- 1/2 particles. At the critical threshold p = n1/2 we find
a sharp classical-quantum phase transition between the normal distribution and
the Wigner semicircle law. The former is characteristic to classical systems with
commuting variables, while the latter is a signature of noncommutative random
matrix theory.
author:
- 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
last_name: Schröder
citation:
ama: Erdös L, Schröder DJ. Phase transition in the density of states of quantum
spin glasses. Mathematical Physics, Analysis and Geometry. 2014;17(3-4):441-464.
doi:10.1007/s11040-014-9164-3
apa: Erdös, L., & Schröder, D. J. (2014). Phase transition in the density of
states of quantum spin glasses. Mathematical Physics, Analysis and Geometry.
Springer. https://doi.org/10.1007/s11040-014-9164-3
chicago: Erdös, László, and Dominik J Schröder. “Phase Transition in the Density
of States of Quantum Spin Glasses.” Mathematical Physics, Analysis and Geometry.
Springer, 2014. https://doi.org/10.1007/s11040-014-9164-3.
ieee: L. Erdös and D. J. Schröder, “Phase transition in the density of states of
quantum spin glasses,” Mathematical Physics, Analysis and Geometry, vol.
17, no. 3–4. Springer, pp. 441–464, 2014.
ista: Erdös L, Schröder DJ. 2014. Phase transition in the density of states of quantum
spin glasses. Mathematical Physics, Analysis and Geometry. 17(3–4), 441–464.
mla: Erdös, László, and Dominik J. Schröder. “Phase Transition in the Density of
States of Quantum Spin Glasses.” Mathematical Physics, Analysis and Geometry,
vol. 17, no. 3–4, Springer, 2014, pp. 441–64, doi:10.1007/s11040-014-9164-3.
short: L. Erdös, D.J. Schröder, Mathematical Physics, Analysis and Geometry 17 (2014)
441–464.
date_created: 2018-12-11T11:55:15Z
date_published: 2014-12-17T00:00:00Z
date_updated: 2021-01-12T06:54:45Z
day: '17'
department:
- _id: LaEr
doi: 10.1007/s11040-014-9164-3
ec_funded: 1
intvolume: ' 17'
issue: 3-4
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1407.1552
month: '12'
oa: 1
oa_version: Submitted Version
page: 441 - 464
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '338804'
name: Random matrices, universality and disordered quantum systems
publication: Mathematical Physics, Analysis and Geometry
publication_status: published
publisher: Springer
publist_id: '5053'
quality_controlled: '1'
scopus_import: 1
status: public
title: Phase transition in the density of states of quantum spin glasses
type: journal_article
user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87
volume: 17
year: '2014'
...
---
_id: '2179'
abstract:
- lang: eng
text: We extend the proof of the local semicircle law for generalized Wigner matrices
given in MR3068390 to the case when the matrix of variances has an eigenvalue
-1. In particular, this result provides a short proof of the optimal local Marchenko-Pastur
law at the hard edge (i.e. around zero) for sample covariance matrices X*X, where
the variances of the entries of X may vary.
author:
- first_name: Oskari H
full_name: Ajanki, Oskari H
id: 36F2FB7E-F248-11E8-B48F-1D18A9856A87
last_name: Ajanki
- first_name: László
full_name: Erdös, László
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
- first_name: Torben H
full_name: Krüger, Torben H
id: 3020C786-F248-11E8-B48F-1D18A9856A87
last_name: Krüger
orcid: 0000-0002-4821-3297
citation:
ama: Ajanki OH, Erdös L, Krüger TH. Local semicircle law with imprimitive variance
matrix. Electronic Communications in Probability. 2014;19. doi:10.1214/ECP.v19-3121
apa: Ajanki, O. H., Erdös, L., & Krüger, T. H. (2014). Local semicircle law
with imprimitive variance matrix. Electronic Communications in Probability.
Institute of Mathematical Statistics. https://doi.org/10.1214/ECP.v19-3121
chicago: Ajanki, Oskari H, László Erdös, and Torben H Krüger. “Local Semicircle
Law with Imprimitive Variance Matrix.” Electronic Communications in Probability.
Institute of Mathematical Statistics, 2014. https://doi.org/10.1214/ECP.v19-3121.
ieee: O. H. Ajanki, L. Erdös, and T. H. Krüger, “Local semicircle law with imprimitive
variance matrix,” Electronic Communications in Probability, vol. 19. Institute
of Mathematical Statistics, 2014.
ista: Ajanki OH, Erdös L, Krüger TH. 2014. Local semicircle law with imprimitive
variance matrix. Electronic Communications in Probability. 19.
mla: Ajanki, Oskari H., et al. “Local Semicircle Law with Imprimitive Variance Matrix.”
Electronic Communications in Probability, vol. 19, Institute of Mathematical
Statistics, 2014, doi:10.1214/ECP.v19-3121.
short: O.H. Ajanki, L. Erdös, T.H. Krüger, Electronic Communications in Probability
19 (2014).
date_created: 2018-12-11T11:56:10Z
date_published: 2014-06-09T00:00:00Z
date_updated: 2021-01-12T06:55:48Z
day: '09'
ddc:
- '570'
department:
- _id: LaEr
doi: 10.1214/ECP.v19-3121
file:
- access_level: open_access
checksum: bd8a041c76d62fe820bf73ff13ce7d1b
content_type: application/pdf
creator: system
date_created: 2018-12-12T10:09:06Z
date_updated: 2020-07-14T12:45:31Z
file_id: '4729'
file_name: IST-2016-426-v1+1_3121-17518-1-PB.pdf
file_size: 327322
relation: main_file
file_date_updated: 2020-07-14T12:45:31Z
has_accepted_license: '1'
intvolume: ' 19'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
publication: Electronic Communications in Probability
publication_status: published
publisher: Institute of Mathematical Statistics
publist_id: '4803'
pubrep_id: '426'
quality_controlled: '1'
scopus_import: 1
status: public
title: Local semicircle law with imprimitive variance matrix
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: 4435EBFC-F248-11E8-B48F-1D18A9856A87
volume: 19
year: '2014'
...
---
_id: '2225'
abstract:
- lang: eng
text: "We consider sample covariance matrices of the form X∗X, where X is an M×N
matrix with independent random entries. We prove the isotropic local Marchenko-Pastur
law, i.e. we prove that the resolvent (X∗X−z)−1 converges to a multiple of the
identity in the sense of quadratic forms. More precisely, we establish sharp high-probability
bounds on the quantity ⟨v,(X∗X−z)−1w⟩−⟨v,w⟩m(z), where m is the Stieltjes transform
of the Marchenko-Pastur law and v,w∈CN. We require the logarithms of the dimensions
M and N to be comparable. Our result holds down to scales Iz≥N−1+ε and throughout
the entire spectrum away from 0. We also prove analogous results for generalized
Wigner matrices.\r\n"
article_number: '33'
author:
- first_name: Alex
full_name: Bloemendal, Alex
last_name: Bloemendal
- first_name: László
full_name: Erdös, László
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
- first_name: Antti
full_name: Knowles, Antti
last_name: Knowles
- first_name: Horng
full_name: Yau, Horng
last_name: Yau
- first_name: Jun
full_name: Yin, Jun
last_name: Yin
citation:
ama: Bloemendal A, Erdös L, Knowles A, Yau H, Yin J. Isotropic local laws for sample
covariance and generalized Wigner matrices. Electronic Journal of Probability.
2014;19. doi:10.1214/EJP.v19-3054
apa: Bloemendal, A., Erdös, L., Knowles, A., Yau, H., & Yin, J. (2014). Isotropic
local laws for sample covariance and generalized Wigner matrices. Electronic
Journal of Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/EJP.v19-3054
chicago: Bloemendal, Alex, László Erdös, Antti Knowles, Horng Yau, and Jun Yin.
“Isotropic Local Laws for Sample Covariance and Generalized Wigner Matrices.”
Electronic Journal of Probability. Institute of Mathematical Statistics,
2014. https://doi.org/10.1214/EJP.v19-3054.
ieee: A. Bloemendal, L. Erdös, A. Knowles, H. Yau, and J. Yin, “Isotropic local
laws for sample covariance and generalized Wigner matrices,” Electronic Journal
of Probability, vol. 19. Institute of Mathematical Statistics, 2014.
ista: Bloemendal A, Erdös L, Knowles A, Yau H, Yin J. 2014. Isotropic local laws
for sample covariance and generalized Wigner matrices. Electronic Journal of Probability.
19, 33.
mla: Bloemendal, Alex, et al. “Isotropic Local Laws for Sample Covariance and Generalized
Wigner Matrices.” Electronic Journal of Probability, vol. 19, 33, Institute
of Mathematical Statistics, 2014, doi:10.1214/EJP.v19-3054.
short: A. Bloemendal, L. Erdös, A. Knowles, H. Yau, J. Yin, Electronic Journal of
Probability 19 (2014).
date_created: 2018-12-11T11:56:25Z
date_published: 2014-03-15T00:00:00Z
date_updated: 2021-01-12T06:56:07Z
day: '15'
ddc:
- '510'
department:
- _id: LaEr
doi: 10.1214/EJP.v19-3054
ec_funded: 1
file:
- access_level: open_access
checksum: 7eb297ff367a2ee73b21b6dd1e1948e4
content_type: application/pdf
creator: system
date_created: 2018-12-12T10:14:06Z
date_updated: 2020-07-14T12:45:34Z
file_id: '5055'
file_name: IST-2016-427-v1+1_3054-16624-4-PB.pdf
file_size: 810150
relation: main_file
file_date_updated: 2020-07-14T12:45:34Z
has_accepted_license: '1'
intvolume: ' 19'
language:
- iso: eng
month: '03'
oa: 1
oa_version: Published Version
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '338804'
name: Random matrices, universality and disordered quantum systems
publication: Electronic Journal of Probability
publication_identifier:
issn:
- '10836489'
publication_status: published
publisher: Institute of Mathematical Statistics
publist_id: '4739'
pubrep_id: '427'
quality_controlled: '1'
status: public
title: Isotropic local laws for sample covariance and generalized 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: 19
year: '2014'
...
---
_id: '1507'
abstract:
- lang: eng
text: The Wigner-Dyson-Gaudin-Mehta conjecture asserts that the local eigenvalue
statistics of large real and complex Hermitian matrices with independent, identically
distributed entries are universal in a sense that they depend only on the symmetry
class of the matrix and otherwise are independent of the details of the distribution.
We present the recent solution to this half-century old conjecture. We explain
how stochastic tools, such as the Dyson Brownian motion, and PDE ideas, such as
De Giorgi-Nash-Moser regularity theory, were combined in the solution. We also
show related results for log-gases that represent a universal model for strongly
correlated systems. Finally, in the spirit of Wigner’s original vision, we discuss
the extensions of these universality results to more realistic physical systems
such as random band matrices.
acknowledgement: The author is partially supported by SFB-TR 12 Grant of the German
Research Council.
article_processing_charge: No
author:
- first_name: László
full_name: Erdös, László
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
citation:
ama: 'Erdös L. Random matrices, log-gases and Hölder regularity. In: Proceedings
of the International Congress of Mathematicians. Vol 3. International Congress
of Mathematicians; 2014:214-236.'
apa: 'Erdös, L. (2014). Random matrices, log-gases and Hölder regularity. In Proceedings
of the International Congress of Mathematicians (Vol. 3, pp. 214–236). Seoul,
Korea: International Congress of Mathematicians.'
chicago: Erdös, László. “Random Matrices, Log-Gases and Hölder Regularity.” In Proceedings
of the International Congress of Mathematicians, 3:214–36. International Congress
of Mathematicians, 2014.
ieee: L. Erdös, “Random matrices, log-gases and Hölder regularity,” in Proceedings
of the International Congress of Mathematicians, Seoul, Korea, 2014, vol.
3, pp. 214–236.
ista: 'Erdös L. 2014. Random matrices, log-gases and Hölder regularity. Proceedings
of the International Congress of Mathematicians. ICM: International Congress of
Mathematicians vol. 3, 214–236.'
mla: Erdös, László. “Random Matrices, Log-Gases and Hölder Regularity.” Proceedings
of the International Congress of Mathematicians, vol. 3, International Congress
of Mathematicians, 2014, pp. 214–36.
short: L. Erdös, in:, Proceedings of the International Congress of Mathematicians,
International Congress of Mathematicians, 2014, pp. 214–236.
conference:
end_date: 2014-08-21
location: Seoul, Korea
name: 'ICM: International Congress of Mathematicians'
start_date: 2014-08-13
date_created: 2018-12-11T11:52:25Z
date_published: 2014-08-01T00:00:00Z
date_updated: 2023-10-17T11:12:55Z
day: '01'
department:
- _id: LaEr
ec_funded: 1
intvolume: ' 3'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1407.5752
month: '08'
oa: 1
oa_version: Submitted Version
page: 214 - 236
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '338804'
name: Random matrices, universality and disordered quantum systems
publication: Proceedings of the International Congress of Mathematicians
publication_status: published
publisher: International Congress of Mathematicians
publist_id: '5670'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Random matrices, log-gases and Hölder regularity
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 3
year: '2014'
...
---
_id: '2698'
abstract:
- lang: eng
text: We consider non-interacting particles subject to a fixed external potential
V and a self-generated magnetic field B. The total energy includes the field energy
β∫B2 and we minimize over all particle states and magnetic fields. In the case
of spin-1/2 particles this minimization leads to the coupled Maxwell-Pauli system.
The parameter β tunes the coupling strength between the field and the particles
and it effectively determines the strength of the field. We investigate the stability
and the semiclassical asymptotics, h→0, of the total ground state energy E(β,h,V).
The relevant parameter measuring the field strength in the semiclassical limit
is κ=βh. We are not able to give the exact leading order semiclassical asymptotics
uniformly in κ or even for fixed κ. We do however give upper and lower bounds
on E with almost matching dependence on κ. In the simultaneous limit h→0 and κ→∞
we show that the standard non-magnetic Weyl asymptotics holds. The same result
also holds for the spinless case, i.e. where the Pauli operator is replaced by
the Schrödinger operator.
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: Søren
full_name: Fournais, Søren
last_name: Fournais
- first_name: Jan
full_name: Solovej, Jan
last_name: Solovej
citation:
ama: Erdös L, Fournais S, Solovej J. Stability and semiclassics in self-generated
fields. Journal of the European Mathematical Society. 2013;15(6):2093-2113.
doi:10.4171/JEMS/416
apa: Erdös, L., Fournais, S., & Solovej, J. (2013). Stability and semiclassics
in self-generated fields. Journal of the European Mathematical Society.
European Mathematical Society. https://doi.org/10.4171/JEMS/416
chicago: Erdös, László, Søren Fournais, and Jan Solovej. “Stability and Semiclassics
in Self-Generated Fields.” Journal of the European Mathematical Society.
European Mathematical Society, 2013. https://doi.org/10.4171/JEMS/416.
ieee: L. Erdös, S. Fournais, and J. Solovej, “Stability and semiclassics in self-generated
fields,” Journal of the European Mathematical Society, vol. 15, no. 6.
European Mathematical Society, pp. 2093–2113, 2013.
ista: Erdös L, Fournais S, Solovej J. 2013. Stability and semiclassics in self-generated
fields. Journal of the European Mathematical Society. 15(6), 2093–2113.
mla: Erdös, László, et al. “Stability and Semiclassics in Self-Generated Fields.”
Journal of the European Mathematical Society, vol. 15, no. 6, European
Mathematical Society, 2013, pp. 2093–113, doi:10.4171/JEMS/416.
short: L. Erdös, S. Fournais, J. Solovej, Journal of the European Mathematical Society
15 (2013) 2093–2113.
date_created: 2018-12-11T11:59:07Z
date_published: 2013-10-16T00:00:00Z
date_updated: 2021-01-12T06:59:07Z
day: '16'
department:
- _id: LaEr
doi: 10.4171/JEMS/416
external_id:
arxiv:
- '1105.0506'
intvolume: ' 15'
issue: '6'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1105.0506
month: '10'
oa: 1
oa_version: Preprint
page: 2093 - 2113
publication: Journal of the European Mathematical Society
publication_status: published
publisher: European Mathematical Society
publist_id: '4198'
quality_controlled: '1'
status: public
title: Stability and semiclassics in self-generated fields
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 15
year: '2013'
...