---
_id: '10405'
abstract:
- lang: eng
text: 'We consider large non-Hermitian random matrices X with complex, independent,
identically distributed centred entries and show that the linear statistics of
their eigenvalues are asymptotically Gaussian for test functions having 2+ϵ derivatives.
Previously this result was known only for a few special cases; either the test
functions were required to be analytic [72], or the distribution of the matrix
elements needed to be Gaussian [73], or at least match the Gaussian up to the
first four moments [82, 56]. We find the exact dependence of the limiting variance
on the fourth cumulant that was not known before. The proof relies on two novel
ingredients: (i) a local law for a product of two resolvents of the Hermitisation
of X with different spectral parameters and (ii) a coupling of several weakly
dependent Dyson Brownian motions. These methods are also the key inputs for our
analogous results on the linear eigenvalue statistics of real matrices X that
are presented in the companion paper [32]. '
acknowledgement: L.E. would like to thank Nathanaël Berestycki and D.S.would like
to thank Nina Holden for valuable discussions on the Gaussian freefield.G.C. and
L.E. are partially supported by ERC Advanced Grant No. 338804.G.C. received funding
from the European Union’s Horizon 2020 research and in-novation programme under
the Marie Skłodowska-Curie Grant Agreement No.665385. D.S. is supported by Dr. Max
Rössler, the Walter Haefner Foundation, and the ETH Zürich Foundation.
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Giorgio
full_name: Cipolloni, Giorgio
id: 42198EFA-F248-11E8-B48F-1D18A9856A87
last_name: Cipolloni
orcid: 0000-0002-4901-7992
- first_name: László
full_name: Erdös, László
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
- first_name: Dominik J
full_name: Schröder, Dominik J
id: 408ED176-F248-11E8-B48F-1D18A9856A87
last_name: Schröder
orcid: 0000-0002-2904-1856
citation:
ama: Cipolloni G, Erdös L, Schröder DJ. Central limit theorem for linear eigenvalue
statistics of non-Hermitian random matrices. Communications on Pure and Applied
Mathematics. 2023;76(5):946-1034. doi:10.1002/cpa.22028
apa: Cipolloni, G., Erdös, L., & Schröder, D. J. (2023). Central limit theorem
for linear eigenvalue statistics of non-Hermitian random matrices. Communications
on Pure and Applied Mathematics. Wiley. https://doi.org/10.1002/cpa.22028
chicago: Cipolloni, Giorgio, László Erdös, and Dominik J Schröder. “Central Limit
Theorem for Linear Eigenvalue Statistics of Non-Hermitian Random Matrices.” Communications
on Pure and Applied Mathematics. Wiley, 2023. https://doi.org/10.1002/cpa.22028.
ieee: G. Cipolloni, L. Erdös, and D. J. Schröder, “Central limit theorem for linear
eigenvalue statistics of non-Hermitian random matrices,” Communications on
Pure and Applied Mathematics, vol. 76, no. 5. Wiley, pp. 946–1034, 2023.
ista: Cipolloni G, Erdös L, Schröder DJ. 2023. Central limit theorem for linear
eigenvalue statistics of non-Hermitian random matrices. Communications on Pure
and Applied Mathematics. 76(5), 946–1034.
mla: Cipolloni, Giorgio, et al. “Central Limit Theorem for Linear Eigenvalue Statistics
of Non-Hermitian Random Matrices.” Communications on Pure and Applied Mathematics,
vol. 76, no. 5, Wiley, 2023, pp. 946–1034, doi:10.1002/cpa.22028.
short: G. Cipolloni, L. Erdös, D.J. Schröder, Communications on Pure and Applied
Mathematics 76 (2023) 946–1034.
date_created: 2021-12-05T23:01:41Z
date_published: 2023-05-01T00:00:00Z
date_updated: 2023-10-04T09:22:55Z
day: '01'
ddc:
- '510'
department:
- _id: LaEr
doi: 10.1002/cpa.22028
ec_funded: 1
external_id:
arxiv:
- '1912.04100'
isi:
- '000724652500001'
file:
- access_level: open_access
checksum: 8346bc2642afb4ccb7f38979f41df5d9
content_type: application/pdf
creator: dernst
date_created: 2023-10-04T09:21:48Z
date_updated: 2023-10-04T09:21:48Z
file_id: '14388'
file_name: 2023_CommPureMathematics_Cipolloni.pdf
file_size: 803440
relation: main_file
success: 1
file_date_updated: 2023-10-04T09:21:48Z
has_accepted_license: '1'
intvolume: ' 76'
isi: 1
issue: '5'
language:
- iso: eng
month: '05'
oa: 1
oa_version: Published Version
page: 946-1034
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '338804'
name: Random matrices, universality and disordered quantum systems
- _id: 2564DBCA-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '665385'
name: International IST Doctoral Program
publication: Communications on Pure and Applied Mathematics
publication_identifier:
eissn:
- 1097-0312
issn:
- 0010-3640
publication_status: published
publisher: Wiley
quality_controlled: '1'
scopus_import: '1'
status: public
title: Central limit theorem for linear eigenvalue statistics of non-Hermitian random
matrices
tmp:
image: /images/cc_by_nc_nd.png
legal_code_url: https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode
name: Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International
(CC BY-NC-ND 4.0)
short: CC BY-NC-ND (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 76
year: '2023'
...
---
_id: '8500'
abstract:
- lang: eng
text: The main model studied in this paper is a lattice of pendula with a nearest‐neighbor
coupling. If the coupling is weak, then the system is near‐integrable and KAM
tori fill most of the phase space. For all KAM trajectories the energy of each
pendulum stays within a narrow band for all time. Still, we show that for an arbitrarily
weak coupling of a certain localized type, the neighboring pendula can exchange
energy. In fact, the energy can be transferred between the pendula in any prescribed
way.
article_processing_charge: No
article_type: original
author:
- first_name: Vadim
full_name: Kaloshin, Vadim
id: FE553552-CDE8-11E9-B324-C0EBE5697425
last_name: Kaloshin
orcid: 0000-0002-6051-2628
- first_name: Mark
full_name: Levi, Mark
last_name: Levi
- first_name: Maria
full_name: Saprykina, Maria
last_name: Saprykina
citation:
ama: Kaloshin V, Levi M, Saprykina M. Arnol′d diffusion in a pendulum lattice. Communications
on Pure and Applied Mathematics. 2014;67(5):748-775. doi:10.1002/cpa.21509
apa: Kaloshin, V., Levi, M., & Saprykina, M. (2014). Arnol′d diffusion in a
pendulum lattice. Communications on Pure and Applied Mathematics. Wiley.
https://doi.org/10.1002/cpa.21509
chicago: Kaloshin, Vadim, Mark Levi, and Maria Saprykina. “Arnol′d Diffusion in
a Pendulum Lattice.” Communications on Pure and Applied Mathematics. Wiley,
2014. https://doi.org/10.1002/cpa.21509.
ieee: V. Kaloshin, M. Levi, and M. Saprykina, “Arnol′d diffusion in a pendulum lattice,”
Communications on Pure and Applied Mathematics, vol. 67, no. 5. Wiley,
pp. 748–775, 2014.
ista: Kaloshin V, Levi M, Saprykina M. 2014. Arnol′d diffusion in a pendulum lattice.
Communications on Pure and Applied Mathematics. 67(5), 748–775.
mla: Kaloshin, Vadim, et al. “Arnol′d Diffusion in a Pendulum Lattice.” Communications
on Pure and Applied Mathematics, vol. 67, no. 5, Wiley, 2014, pp. 748–75,
doi:10.1002/cpa.21509.
short: V. Kaloshin, M. Levi, M. Saprykina, Communications on Pure and Applied Mathematics
67 (2014) 748–775.
date_created: 2020-09-18T10:47:01Z
date_published: 2014-05-01T00:00:00Z
date_updated: 2022-08-25T13:58:13Z
day: '01'
doi: 10.1002/cpa.21509
extern: '1'
intvolume: ' 67'
issue: '5'
keyword:
- Applied Mathematics
- General Mathematics
language:
- iso: eng
month: '05'
oa_version: None
page: 748-775
publication: Communications on Pure and Applied Mathematics
publication_identifier:
issn:
- 0010-3640
publication_status: published
publisher: Wiley
quality_controlled: '1'
status: public
title: Arnol′d diffusion in a pendulum lattice
type: journal_article
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 67
year: '2014'
...
---
_id: '8517'
abstract:
- lang: eng
text: We consider the evolution of a connected set on the plane carried by a space
periodic incompressible stochastic flow. While for almost every realization of
the stochastic flow at time t most of the particles are at a distance of order
equation image away from the origin, there is a measure zero set of points that
escape to infinity at the linear rate. We study the set of points visited by the
original set by time t and show that such a set, when scaled down by the factor
of t, has a limiting nonrandom shape.
article_processing_charge: No
article_type: original
author:
- first_name: Dmitry
full_name: Dolgopyat, Dmitry
last_name: Dolgopyat
- first_name: Vadim
full_name: Kaloshin, Vadim
id: FE553552-CDE8-11E9-B324-C0EBE5697425
last_name: Kaloshin
orcid: 0000-0002-6051-2628
- first_name: Leonid
full_name: Koralov, Leonid
last_name: Koralov
citation:
ama: Dolgopyat D, Kaloshin V, Koralov L. A limit shape theorem for periodic stochastic
dispersion. Communications on Pure and Applied Mathematics. 2004;57(9):1127-1158.
doi:10.1002/cpa.20032
apa: Dolgopyat, D., Kaloshin, V., & Koralov, L. (2004). A limit shape theorem
for periodic stochastic dispersion. Communications on Pure and Applied Mathematics.
Wiley. https://doi.org/10.1002/cpa.20032
chicago: Dolgopyat, Dmitry, Vadim Kaloshin, and Leonid Koralov. “A Limit Shape Theorem
for Periodic Stochastic Dispersion.” Communications on Pure and Applied Mathematics.
Wiley, 2004. https://doi.org/10.1002/cpa.20032.
ieee: D. Dolgopyat, V. Kaloshin, and L. Koralov, “A limit shape theorem for periodic
stochastic dispersion,” Communications on Pure and Applied Mathematics,
vol. 57, no. 9. Wiley, pp. 1127–1158, 2004.
ista: Dolgopyat D, Kaloshin V, Koralov L. 2004. A limit shape theorem for periodic
stochastic dispersion. Communications on Pure and Applied Mathematics. 57(9),
1127–1158.
mla: Dolgopyat, Dmitry, et al. “A Limit Shape Theorem for Periodic Stochastic Dispersion.”
Communications on Pure and Applied Mathematics, vol. 57, no. 9, Wiley,
2004, pp. 1127–58, doi:10.1002/cpa.20032.
short: D. Dolgopyat, V. Kaloshin, L. Koralov, Communications on Pure and Applied
Mathematics 57 (2004) 1127–1158.
date_created: 2020-09-18T10:49:12Z
date_published: 2004-09-01T00:00:00Z
date_updated: 2021-01-12T08:19:50Z
day: '01'
doi: 10.1002/cpa.20032
extern: '1'
intvolume: ' 57'
issue: '9'
keyword:
- Applied Mathematics
- General Mathematics
language:
- iso: eng
month: '09'
oa_version: None
page: 1127-1158
publication: Communications on Pure and Applied Mathematics
publication_identifier:
issn:
- 0010-3640
- 1097-0312
publication_status: published
publisher: Wiley
quality_controlled: '1'
status: public
title: A limit shape theorem for periodic stochastic dispersion
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 57
year: '2004'
...
---
_id: '2731'
abstract:
- lang: eng
text: We study the time evolution of a quantum particle in a Gaussian random environment.
We show that in the weak coupling limit the Wigner distribution of the wave function
converges to a solution of a linear Boltzmann equation globally in time. The Boltzmann
collision kernel is given by the Born approximation of the quantum differential
scattering cross section.
acknowledgement: Partially supported by U.S. National Science Foundation grants DMS-9403462,
9703752. We would like to thank H. Spohn for his several comments and discussions
on this project. Part of this work was done during the time when L. E. visited the
Erwin Schrödinger Institute in Vienna and when both authors visited the Center of
Theoretical Sciences in Taiwan. We thank them for the hospitality and the support
of this work.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: László
full_name: Erdös, László
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
- first_name: Horng
full_name: Yau, Horng
last_name: Yau
citation:
ama: Erdös L, Yau H. Linear Boltzmann equation as the weak coupling limit of a random
Schrödinger equation. Communications on Pure and Applied Mathematics. 2000;53(6):667-735.
doi:10.1002/(SICI)1097-0312(200006)53:6<667::AID-CPA1>3.0.CO;2-5
apa: Erdös, L., & Yau, H. (2000). Linear Boltzmann equation as the weak coupling
limit of a random Schrödinger equation. Communications on Pure and Applied
Mathematics. Wiley-Blackwell. https://doi.org/10.1002/(SICI)1097-0312(200006)53:6<667::AID-CPA1>3.0.CO;2-5
chicago: Erdös, László, and Horng Yau. “Linear Boltzmann Equation as the Weak Coupling
Limit of a Random Schrödinger Equation.” Communications on Pure and Applied
Mathematics. Wiley-Blackwell, 2000. https://doi.org/10.1002/(SICI)1097-0312(200006)53:6<667::AID-CPA1>3.0.CO;2-5.
ieee: L. Erdös and H. Yau, “Linear Boltzmann equation as the weak coupling limit
of a random Schrödinger equation,” Communications on Pure and Applied Mathematics,
vol. 53, no. 6. Wiley-Blackwell, pp. 667–735, 2000.
ista: Erdös L, Yau H. 2000. Linear Boltzmann equation as the weak coupling limit
of a random Schrödinger equation. Communications on Pure and Applied Mathematics.
53(6), 667–735.
mla: Erdös, László, and Horng Yau. “Linear Boltzmann Equation as the Weak Coupling
Limit of a Random Schrödinger Equation.” Communications on Pure and Applied
Mathematics, vol. 53, no. 6, Wiley-Blackwell, 2000, pp. 667–735, doi:10.1002/(SICI)1097-0312(200006)53:6<667::AID-CPA1>3.0.CO;2-5.
short: L. Erdös, H. Yau, Communications on Pure and Applied Mathematics 53 (2000)
667–735.
date_created: 2018-12-11T11:59:18Z
date_published: 2000-06-01T00:00:00Z
date_updated: 2023-05-03T09:09:04Z
day: '01'
doi: 10.1002/(SICI)1097-0312(200006)53:6<667::AID-CPA1>3.0.CO;2-5
extern: '1'
external_id:
arxiv:
- math-ph/9901020
intvolume: ' 53'
issue: '6'
language:
- iso: eng
month: '06'
oa_version: Preprint
page: 667 - 735
publication: Communications on Pure and Applied Mathematics
publication_identifier:
issn:
- 0010-3640
publication_status: published
publisher: Wiley-Blackwell
publist_id: '4161'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Linear Boltzmann equation as the weak coupling limit of a random Schrödinger
equation
type: journal_article
user_id: ea97e931-d5af-11eb-85d4-e6957dddbf17
volume: 53
year: '2000'
...