---
OA_place: repository
OA_type: green
_id: '21933'
abstract:
- lang: eng
text: We consider the liquid drop model with a positive background density in the
thermodynamic limit. We prove a two-term asymptotics for the ground state energy
per unit volume in the dilute limit. Our proof justifies the expectation that
optimal configurations consist of droplets of unit size that arrange themselves
according to minimizers for the Jellium problem for point particles. In particular,
we provide the first rigorous derivation of what is known as the gnocchi phase
in astrophysics.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Rupert L.
full_name: Frank, Rupert L.
last_name: Frank
- first_name: Mathieu
full_name: Lewin, Mathieu
last_name: Lewin
- first_name: Robert
full_name: Seiringer, Robert
id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
last_name: Seiringer
orcid: 0000-0002-6781-0521
citation:
ama: Frank RL, Lewin M, Seiringer R. Liquid drop model for nuclear matter in the
low density limit. Communications on Pure and Applied Mathematics. 2026.
doi:10.1002/cpa.70039
apa: Frank, R. L., Lewin, M., & Seiringer, R. (2026). Liquid drop model for
nuclear matter in the low density limit. Communications on Pure and Applied
Mathematics. Wiley. https://doi.org/10.1002/cpa.70039
chicago: Frank, Rupert L., Mathieu Lewin, and Robert Seiringer. “Liquid Drop Model
for Nuclear Matter in the Low Density Limit.” Communications on Pure and Applied
Mathematics. Wiley, 2026. https://doi.org/10.1002/cpa.70039.
ieee: R. L. Frank, M. Lewin, and R. Seiringer, “Liquid drop model for nuclear matter
in the low density limit,” Communications on Pure and Applied Mathematics.
Wiley, 2026.
ista: Frank RL, Lewin M, Seiringer R. 2026. Liquid drop model for nuclear matter
in the low density limit. Communications on Pure and Applied Mathematics.
mla: Frank, Rupert L., et al. “Liquid Drop Model for Nuclear Matter in the Low Density
Limit.” Communications on Pure and Applied Mathematics, Wiley, 2026, doi:10.1002/cpa.70039.
short: R.L. Frank, M. Lewin, R. Seiringer, Communications on Pure and Applied Mathematics
(2026).
date_created: 2026-05-31T22:02:13Z
date_published: 2026-01-01T00:00:00Z
date_updated: 2026-06-02T06:58:54Z
day: '01'
department:
- _id: RoSe
doi: 10.1002/cpa.70039
external_id:
arxiv:
- '2507.14012'
fulldoi: https://doi.org/10.1002/cpa.70039
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2507.14012
month: '01'
oa: 1
oa_version: Preprint
publication: Communications on Pure and Applied Mathematics
publication_identifier:
eissn:
- 1097-0312
issn:
- 0010-3640
publication_status: epub_ahead
publisher: Wiley
scopus_import: '1'
status: public
title: Liquid drop model for nuclear matter in the low density limit
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2026'
...
---
OA_place: publisher
OA_type: hybrid
_id: '21472'
abstract:
- lang: eng
text: We study the ground state energy of a gas of spin 1/2 fermions with repulsive
short-range interactions. We derive an upper bound that agrees, at low density
e, with the Huang–Yang conjecture. The latter captures the first three terms in
an asymptotic low-density expansion, and in particular the Huang–Yang correction
term of order e^7/3. Our trial state is constructed using an adaptation of the
bosonic Bogoliubov theory to the Fermi system, where the correlation structure
of fermionic particles is incorporated by quasi-bosonic Bogoliubov transformations.
In the latter, it is important to consider a modified zero-energy scattering equation
that takes into account the presence of the Fermi sea, in the spirit of the Bethe–Goldstone
equation.
acknowledgement: "We thank the referees for valuable remarks. This work was partially
funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)
via the TRR 352 – Project-ID 470903074. PTN was partially supported by the European
Research Council via the ERC Consolidator Grant RAMBAS – Project-Nr. 10104424.\r\nOpen
access publishing facilitated by Università degli Studi di Milano, as part of the
Wiley - CRUI-CARE agreement."
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Emanuela L.
full_name: Giacomelli, Emanuela L.
last_name: Giacomelli
- first_name: Christian
full_name: Hainzl, Christian
last_name: Hainzl
- first_name: Phan Thành
full_name: Nam, Phan Thành
last_name: Nam
- first_name: Robert
full_name: Seiringer, Robert
id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
last_name: Seiringer
orcid: 0000-0002-6781-0521
citation:
ama: 'Giacomelli EL, Hainzl C, Nam PT, Seiringer R. The Huang–Yang formula for the
low-density Fermi gas: Upper bound. Communications on Pure and Applied Mathematics.
2026;79(8):1919-1972. doi:10.1002/cpa.70040'
apa: 'Giacomelli, E. L., Hainzl, C., Nam, P. T., & Seiringer, R. (2026). The
Huang–Yang formula for the low-density Fermi gas: Upper bound. Communications
on Pure and Applied Mathematics. Wiley. https://doi.org/10.1002/cpa.70040'
chicago: 'Giacomelli, Emanuela L., Christian Hainzl, Phan Thành Nam, and Robert
Seiringer. “The Huang–Yang Formula for the Low-Density Fermi Gas: Upper Bound.”
Communications on Pure and Applied Mathematics. Wiley, 2026. https://doi.org/10.1002/cpa.70040.'
ieee: 'E. L. Giacomelli, C. Hainzl, P. T. Nam, and R. Seiringer, “The Huang–Yang
formula for the low-density Fermi gas: Upper bound,” Communications on Pure
and Applied Mathematics, vol. 79, no. 8. Wiley, pp. 1919–1972, 2026.'
ista: 'Giacomelli EL, Hainzl C, Nam PT, Seiringer R. 2026. The Huang–Yang formula
for the low-density Fermi gas: Upper bound. Communications on Pure and Applied
Mathematics. 79(8), 1919–1972.'
mla: 'Giacomelli, Emanuela L., et al. “The Huang–Yang Formula for the Low-Density
Fermi Gas: Upper Bound.” Communications on Pure and Applied Mathematics,
vol. 79, no. 8, Wiley, 2026, pp. 1919–72, doi:10.1002/cpa.70040.'
short: E.L. Giacomelli, C. Hainzl, P.T. Nam, R. Seiringer, Communications on Pure
and Applied Mathematics 79 (2026) 1919–1972.
das_tickbox: '0'
date_created: 2026-03-22T23:04:33Z
date_published: 2026-08-01T00:00:00Z
date_updated: 2026-07-27T11:40:33Z
day: '01'
ddc:
- '510'
department:
- _id: RoSe
doi: 10.1002/cpa.70040
external_id:
arxiv:
- '2409.17914'
file:
- access_level: open_access
checksum: 4ef624157c2f6cb837be229dd6b912cc
content_type: application/pdf
creator: dernst
date_created: 2026-07-27T11:38:57Z
date_updated: 2026-07-27T11:38:57Z
file_id: '22423'
file_name: 2026_CommPureApplMath_Giacomelli.pdf
file_size: 2224756
relation: main_file
success: 1
file_date_updated: 2026-07-27T11:38:57Z
fulldoi: https://doi.org/10.1002/cpa.70040
has_accepted_license: '1'
intvolume: ' 79'
issue: '8'
language:
- iso: eng
month: '08'
oa: 1
oa_version: Published Version
page: 1919-1972
publication: Communications on Pure and Applied Mathematics
publication_identifier:
eissn:
- 1097-0312
issn:
- 0010-3640
publication_status: published
publisher: Wiley
quality_controlled: '1'
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: no
title: 'The Huang–Yang formula for the low-density Fermi gas: Upper bound'
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: 79
year: '2026'
...
---
OA_place: publisher
OA_type: hybrid
_id: '15378'
abstract:
- lang: eng
text: We consider N×N non-Hermitian random matrices of the form X+A, where A is
a general deterministic matrix and N−−√X consists of independent entries with
zero mean, unit variance, and bounded densities. For this ensemble, we prove (i)
a Wegner estimate, i.e. that the local density of eigenvalues is bounded by N1+o(1)
and (ii) that the expected condition number of any bulk eigenvalue is bounded
by N1+o(1); both results are optimal up to the factor No(1). The latter result
complements the very recent matching lower bound obtained in [15] (arXiv:2301.03549)
and improves the N-dependence of the upper bounds in [5,6,32] (arXiv:1906.11819,
arXiv:2005.08930, arXiv:2005.08908). Our main ingredient, a near-optimal lower
tail estimate for the small singular values of X+A−z, is of independent interest.
acknowledgement: László Erdős is partially supported by ERC Advanced Grant “RMTBeyond”
No. 101020331. Hong Chang Ji is supported by ERC Advanced Grant “RMTBeyond” No.
101020331.
article_processing_charge: Yes (via OA deal)
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: Hong Chang
full_name: Ji, Hong Chang
id: dd216c0a-c1f9-11eb-beaf-e9ea9d2de76d
last_name: Ji
citation:
ama: Erdös L, Ji HC. Wegner estimate and upper bound on the eigenvalue condition
number of non-Hermitian random matrices. Communications on Pure and Applied
Mathematics. 2024;77(9):3785-3840. doi:10.1002/cpa.22201
apa: Erdös, L., & Ji, H. C. (2024). Wegner estimate and upper bound on the eigenvalue
condition number of non-Hermitian random matrices. Communications on Pure and
Applied Mathematics. Wiley. https://doi.org/10.1002/cpa.22201
chicago: Erdös, László, and Hong Chang Ji. “Wegner Estimate and Upper Bound on the
Eigenvalue Condition Number of Non-Hermitian Random Matrices.” Communications
on Pure and Applied Mathematics. Wiley, 2024. https://doi.org/10.1002/cpa.22201.
ieee: L. Erdös and H. C. Ji, “Wegner estimate and upper bound on the eigenvalue
condition number of non-Hermitian random matrices,” Communications on Pure
and Applied Mathematics, vol. 77, no. 9. Wiley, pp. 3785–3840, 2024.
ista: Erdös L, Ji HC. 2024. Wegner estimate and upper bound on the eigenvalue condition
number of non-Hermitian random matrices. Communications on Pure and Applied Mathematics.
77(9), 3785–3840.
mla: Erdös, László, and Hong Chang Ji. “Wegner Estimate and Upper Bound on the Eigenvalue
Condition Number of Non-Hermitian Random Matrices.” Communications on Pure
and Applied Mathematics, vol. 77, no. 9, Wiley, 2024, pp. 3785–840, doi:10.1002/cpa.22201.
short: L. Erdös, H.C. Ji, Communications on Pure and Applied Mathematics 77 (2024)
3785–3840.
corr_author: '1'
das_tickbox: '0'
date_created: 2024-05-12T22:01:02Z
date_published: 2024-09-01T00:00:00Z
date_updated: 2026-10-16T08:35:46Z
day: '01'
ddc:
- '510'
department:
- _id: LaEr
doi: 10.1002/cpa.22201
ec_funded: 1
external_id:
arxiv:
- '2301.04981'
isi:
- '001217139900001'
file:
- access_level: open_access
checksum: fbcc9cc7bf274f024e4f4afc9c208f96
content_type: application/pdf
creator: dernst
date_created: 2025-01-09T09:36:41Z
date_updated: 2025-01-09T09:36:41Z
file_id: '18803'
file_name: 2024_CommPureApplMath_Erdoes.pdf
file_size: 566963
relation: main_file
success: 1
file_date_updated: 2025-01-09T09:36:41Z
fulldoi: https://doi.org/10.1002/cpa.22201
has_accepted_license: '1'
intvolume: ' 77'
isi: 1
issue: '9'
language:
- iso: eng
month: '09'
oa: 1
oa_version: Published Version
page: 3785-3840
project:
- _id: 62796744-2b32-11ec-9570-940b20777f1d
call_identifier: H2020
grant_number: '101020331'
name: Random matrices beyond Wigner-Dyson-Mehta
publication: Communications on Pure and Applied Mathematics
publication_identifier:
eissn:
- 1097-0312
issn:
- 0010-3640
publication_status: published
publisher: Wiley
quality_controlled: '1'
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: yes
title: Wegner estimate and upper bound on the eigenvalue condition number 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: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 77
year: '2024'
...
---
_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.
corr_author: '1'
date_created: 2021-12-05T23:01:41Z
date_published: 2023-05-01T00:00:00Z
date_updated: 2025-03-31T16:00:54Z
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
fulldoi: https://doi.org/10.1002/cpa.22028
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: '8603'
abstract:
- lang: eng
text: We consider the Fröhlich polaron model in the strong coupling limit. It is
well‐known that to leading order the ground state energy is given by the (classical)
Pekar energy. In this work, we establish the subleading correction, describing
quantum fluctuation about the classical limit. Our proof applies to a model of
a confined polaron, where both the electron and the polarization field are restricted
to a set of finite volume, with linear size determined by the natural length scale
of the Pekar problem.
acknowledgement: Partial support through National Science Foundation GrantDMS-1363432
(R.L.F.) and the European Research Council (ERC) under the Euro-pean Union’s Horizon
2020 research and innovation programme (grant agreementNo 694227; R.S.), is acknowledged.
Open access funding enabled and organizedby Projekt DEAL.
article_processing_charge: No
article_type: original
author:
- first_name: Rupert
full_name: Frank, Rupert
last_name: Frank
- first_name: Robert
full_name: Seiringer, Robert
id: 4AFD0470-F248-11E8-B48F-1D18A9856A87
last_name: Seiringer
orcid: 0000-0002-6781-0521
citation:
ama: Frank R, Seiringer R. Quantum corrections to the Pekar asymptotics of a strongly
coupled polaron. Communications on Pure and Applied Mathematics. 2021;74(3):544-588.
doi:10.1002/cpa.21944
apa: Frank, R., & Seiringer, R. (2021). Quantum corrections to the Pekar asymptotics
of a strongly coupled polaron. Communications on Pure and Applied Mathematics.
Wiley. https://doi.org/10.1002/cpa.21944
chicago: Frank, Rupert, and Robert Seiringer. “Quantum Corrections to the Pekar
Asymptotics of a Strongly Coupled Polaron.” Communications on Pure and Applied
Mathematics. Wiley, 2021. https://doi.org/10.1002/cpa.21944.
ieee: R. Frank and R. Seiringer, “Quantum corrections to the Pekar asymptotics of
a strongly coupled polaron,” Communications on Pure and Applied Mathematics,
vol. 74, no. 3. Wiley, pp. 544–588, 2021.
ista: Frank R, Seiringer R. 2021. Quantum corrections to the Pekar asymptotics of
a strongly coupled polaron. Communications on Pure and Applied Mathematics. 74(3),
544–588.
mla: Frank, Rupert, and Robert Seiringer. “Quantum Corrections to the Pekar Asymptotics
of a Strongly Coupled Polaron.” Communications on Pure and Applied Mathematics,
vol. 74, no. 3, Wiley, 2021, pp. 544–88, doi:10.1002/cpa.21944.
short: R. Frank, R. Seiringer, Communications on Pure and Applied Mathematics 74
(2021) 544–588.
date_created: 2020-10-04T22:01:37Z
date_published: 2021-03-01T00:00:00Z
date_updated: 2025-07-10T11:57:12Z
day: '01'
ddc:
- '510'
department:
- _id: RoSe
doi: 10.1002/cpa.21944
ec_funded: 1
external_id:
isi:
- '000572991500001'
file:
- access_level: open_access
checksum: 5f665ffa6e6dd958aec5c3040cbcfa84
content_type: application/pdf
creator: dernst
date_created: 2021-03-11T10:03:30Z
date_updated: 2021-03-11T10:03:30Z
file_id: '9236'
file_name: 2021_CommPureApplMath_Frank.pdf
file_size: 334987
relation: main_file
success: 1
file_date_updated: 2021-03-11T10:03:30Z
fulldoi: https://doi.org/10.1002/cpa.21944
has_accepted_license: '1'
intvolume: ' 74'
isi: 1
issue: '3'
language:
- iso: eng
month: '03'
oa: 1
oa_version: Published Version
page: 544-588
project:
- _id: 25C6DC12-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '694227'
name: Analysis of quantum many-body systems
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: Quantum corrections to the Pekar asymptotics of a strongly coupled polaron
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: 74
year: '2021'
...