---
_id: '6511'
abstract:
- lang: eng
text: Let U and V be two independent N by N random matrices that are distributed
according to Haar measure on U(N). Let Σ be a nonnegative deterministic N by N
matrix. The single ring theorem [Ann. of Math. (2) 174 (2011) 1189–1217] asserts
that the empirical eigenvalue distribution of the matrix X:=UΣV∗ converges weakly,
in the limit of large N, to a deterministic measure which is supported on a single
ring centered at the origin in ℂ. Within the bulk regime, that is, in the interior
of the single ring, we establish the convergence of the empirical eigenvalue distribution
on the optimal local scale of order N−1/2+ε and establish the optimal convergence
rate. The same results hold true when U and V are Haar distributed on O(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: László
full_name: Erdös, László
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
- first_name: Kevin
full_name: Schnelli, Kevin
id: 434AD0AE-F248-11E8-B48F-1D18A9856A87
last_name: Schnelli
citation:
ama: Bao Z, Erdös L, Schnelli K. Local single ring theorem on optimal scale. *Annals
of Probability*. 2019;47(3):1270-1334. doi:10.1214/18-AOP1284
apa: Bao, Z., Erdös, L., & Schnelli, K. (2019). Local single ring theorem on
optimal scale. *Annals of Probability*, *47*(3), 1270–1334. https://doi.org/10.1214/18-AOP1284
chicago: 'Bao, Zhigang, László Erdös, and Kevin Schnelli. “Local Single Ring Theorem
on Optimal Scale.” *Annals of Probability* 47, no. 3 (2019): 1270–1334. https://doi.org/10.1214/18-AOP1284.'
ieee: Z. Bao, L. Erdös, and K. Schnelli, “Local single ring theorem on optimal scale,”
*Annals of Probability*, vol. 47, no. 3, pp. 1270–1334, 2019.
ista: Bao Z, Erdös L, Schnelli K. 2019. Local single ring theorem on optimal scale.
Annals of Probability. 47(3), 1270–1334.
mla: Bao, Zhigang, et al. “Local Single Ring Theorem on Optimal Scale.” *Annals
of Probability*, vol. 47, no. 3, Project Euclid, 2019, pp. 1270–334, doi:10.1214/18-AOP1284.
short: Z. Bao, L. Erdös, K. Schnelli, Annals of Probability 47 (2019) 1270–1334.
date_created: 2019-06-02T21:59:13Z
date_published: 2019-05-01T00:00:00Z
date_updated: 2019-08-02T12:39:21Z
day: '01'
department:
- _id: LaEr
doi: 10.1214/18-AOP1284
external_id:
arxiv:
- '1612.05920'
intvolume: ' 47'
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1612.05920
month: '05'
oa: 1
oa_version: Preprint
page: 1270-1334
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
grant_number: '338804'
name: Random matrices, universality and disordered quantum systems
publication: Annals of Probability
publication_identifier:
issn:
- '00911798'
publication_status: published
publisher: Project Euclid
quality_controlled: '1'
status: public
title: Local single ring theorem on optimal scale
type: journal_article
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 47
year: '2019'
...
---
_id: '690'
abstract:
- lang: eng
text: We consider spectral properties and the edge universality of sparse random
matrices, the class of random matrices that includes the adjacency matrices of
the Erdős–Rényi graph model G(N, p). We prove a local law for the eigenvalue density
up to the spectral edges. Under a suitable condition on the sparsity, we also
prove that the rescaled extremal eigenvalues exhibit GOE Tracy–Widom fluctuations
if a deterministic shift of the spectral edge due to the sparsity is included.
For the adjacency matrix of the Erdős–Rényi graph this establishes the Tracy–Widom
fluctuations of the second largest eigenvalue when p is much larger than N−2/3
with a deterministic shift of order (Np)−1.
article_number: 543-616
author:
- first_name: Jii
full_name: Lee, Jii
last_name: Lee
- first_name: Kevin
full_name: Schnelli, Kevin
id: 434AD0AE-F248-11E8-B48F-1D18A9856A87
last_name: Schnelli
citation:
ama: Lee J, Schnelli K. Local law and Tracy–Widom limit for sparse random matrices.
*Probability Theory and Related Fields*. 2018;171(1-2). doi:10.1007/s00440-017-0787-8
apa: Lee, J., & Schnelli, K. (2018). Local law and Tracy–Widom limit for sparse
random matrices. *Probability Theory and Related Fields*, *171*(1–2).
https://doi.org/10.1007/s00440-017-0787-8
chicago: Lee, Jii, and Kevin Schnelli. “Local Law and Tracy–Widom Limit for Sparse
Random Matrices.” *Probability Theory and Related Fields* 171, no. 1–2 (2018).
https://doi.org/10.1007/s00440-017-0787-8.
ieee: J. Lee and K. Schnelli, “Local law and Tracy–Widom limit for sparse random
matrices,” *Probability Theory and Related Fields*, vol. 171, no. 1–2, 2018.
ista: Lee J, Schnelli K. 2018. Local law and Tracy–Widom limit for sparse random
matrices. Probability Theory and Related Fields. 171(1–2).
mla: Lee, Jii, and Kevin Schnelli. “Local Law and Tracy–Widom Limit for Sparse Random
Matrices.” *Probability Theory and Related Fields*, vol. 171, no. 1–2, 543–616,
Springer, 2018, doi:10.1007/s00440-017-0787-8.
short: J. Lee, K. Schnelli, Probability Theory and Related Fields 171 (2018).
date_created: 2018-12-11T11:47:56Z
date_published: 2018-06-14T00:00:00Z
date_updated: 2019-08-02T12:39:27Z
day: '14'
department:
- _id: LaEr
doi: 10.1007/s00440-017-0787-8
external_id:
arxiv:
- '1605.08767'
intvolume: ' 171'
issue: 1-2
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1605.08767
month: '06'
oa: 1
oa_version: Preprint
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
grant_number: '338804'
name: Random matrices, universality and disordered quantum systems
publication: Probability Theory and Related Fields
publication_status: published
publisher: Springer
publist_id: '7017'
quality_controlled: '1'
status: public
title: Local law and Tracy–Widom limit for sparse random matrices
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 171
year: '2018'
...
---
_id: '733'
abstract:
- lang: eng
text: Let A and B be two N by N deterministic Hermitian matrices and let U be an
N by N Haar distributed unitary matrix. It is well known that the spectral distribution
of the sum H = A + UBU∗ converges weakly to the free additive convolution of the
spectral distributions of A and B, as N tends to infinity. We establish the optimal
convergence rate in the bulk of the spectrum.
acknowledgement: Partially supported by ERC Advanced Grant RANMAT No. 338804, Hong
Kong RGC grant ECS 26301517, and the Göran Gustafsson Foundation
author:
- first_name: Zhigang
full_name: Bao, Zhigang
id: 442E6A6C-F248-11E8-B48F-1D18A9856A87
last_name: Bao
orcid: 0000-0003-3036-1475
- first_name: László
full_name: Erdös, László
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
- first_name: Kevin
full_name: Schnelli, Kevin
id: 434AD0AE-F248-11E8-B48F-1D18A9856A87
last_name: Schnelli
citation:
ama: Bao Z, Erdös L, Schnelli K. Convergence rate for spectral distribution of addition
of random matrices. *Advances in Mathematics*. 2017;319:251-291. doi:10.1016/j.aim.2017.08.028
apa: Bao, Z., Erdös, L., & Schnelli, K. (2017). Convergence rate for spectral
distribution of addition of random matrices. *Advances in Mathematics*, *319*,
251–291. https://doi.org/10.1016/j.aim.2017.08.028
chicago: 'Bao, Zhigang, László Erdös, and Kevin Schnelli. “Convergence Rate for
Spectral Distribution of Addition of Random Matrices.” *Advances in Mathematics*
319 (2017): 251–91. https://doi.org/10.1016/j.aim.2017.08.028.'
ieee: Z. Bao, L. Erdös, and K. Schnelli, “Convergence rate for spectral distribution
of addition of random matrices,” *Advances in Mathematics*, vol. 319, pp.
251–291, 2017.
ista: Bao Z, Erdös L, Schnelli K. 2017. Convergence rate for spectral distribution
of addition of random matrices. Advances in Mathematics. 319, 251–291.
mla: Bao, Zhigang, et al. “Convergence Rate for Spectral Distribution of Addition
of Random Matrices.” *Advances in Mathematics*, vol. 319, Academic Press,
2017, pp. 251–91, doi:10.1016/j.aim.2017.08.028.
short: Z. Bao, L. Erdös, K. Schnelli, Advances in Mathematics 319 (2017) 251–291.
date_created: 2018-12-11T11:48:13Z
date_published: 2017-10-15T00:00:00Z
date_updated: 2019-08-02T12:39:29Z
day: '15'
department:
- _id: LaEr
doi: 10.1016/j.aim.2017.08.028
intvolume: ' 319'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1606.03076
month: '10'
oa: 1
oa_version: Submitted Version
page: 251 - 291
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
grant_number: '338804'
name: Random matrices, universality and disordered quantum systems
publication: Advances in Mathematics
publication_status: published
publisher: Academic Press
publist_id: '6935'
quality_controlled: '1'
status: public
title: Convergence rate for spectral distribution of addition of random matrices
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 319
year: '2017'
...
---
_id: '1207'
abstract:
- lang: eng
text: The eigenvalue distribution of the sum of two large Hermitian matrices, when
one of them is conjugated by a Haar distributed unitary matrix, is asymptotically
given by the free convolution of their spectral distributions. We prove that this
convergence also holds locally in the bulk of the spectrum, down to the optimal
scales larger than the eigenvalue spacing. The corresponding eigenvectors are
fully delocalized. Similar results hold for the sum of two real symmetric matrices,
when one is conjugated by Haar orthogonal matrix.
accept: '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: László
full_name: Erdös, László
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
- first_name: Kevin
full_name: Schnelli, Kevin
id: 434AD0AE-F248-11E8-B48F-1D18A9856A87
last_name: Schnelli
cc_license: '''https://creativecommons.org/licenses/by/4.0/'''
citation:
ama: Bao Z, Erdös L, Schnelli K. Local law of addition of random matrices on optimal
scale. *Communications in Mathematical Physics*. 2017;349(3):947-990. doi:10.1007/s00220-016-2805-6
apa: Bao, Z., Erdös, L., & Schnelli, K. (2017). Local law of addition of random
matrices on optimal scale. *Communications in Mathematical Physics*, *349*(3),
947–990. https://doi.org/10.1007/s00220-016-2805-6
chicago: 'Bao, Zhigang, László Erdös, and Kevin Schnelli. “Local Law of Addition
of Random Matrices on Optimal Scale.” *Communications in Mathematical Physics*
349, no. 3 (2017): 947–90. https://doi.org/10.1007/s00220-016-2805-6.'
ieee: Z. Bao, L. Erdös, and K. Schnelli, “Local law of addition of random matrices
on optimal scale,” *Communications in Mathematical Physics*, vol. 349, no.
3, pp. 947–990, 2017.
ista: Bao Z, Erdös L, Schnelli K. 2017. Local law of addition of random matrices
on optimal scale. Communications in Mathematical Physics. 349(3), 947–990.
mla: Bao, Zhigang, et al. “Local Law of Addition of Random Matrices on Optimal Scale.”
*Communications in Mathematical Physics*, vol. 349, no. 3, Springer, 2017,
pp. 947–90, doi:10.1007/s00220-016-2805-6.
short: Z. Bao, L. Erdös, K. Schnelli, Communications in Mathematical Physics 349
(2017) 947–990.
date_created: 2018-12-11T11:50:43Z
date_published: 2017-02-01T00:00:00Z
date_updated: 2019-08-02T12:36:55Z
day: '01'
ddc:
- '530'
department:
- _id: LaEr
doi: 10.1007/s00220-016-2805-6
file:
- access_level: open_access
content_type: application/pdf
creator: system
date_created: 2018-12-12T10:14:47Z
date_updated: 2018-12-12T10:14:47Z
file_id: '5102'
file_name: IST-2016-722-v1+1_s00220-016-2805-6.pdf
file_size: 1033743
open_access: 1
relation: main_file
file_date_updated: 2018-12-12T10:14:47Z
intvolume: ' 349'
issue: '3'
language:
- iso: eng
month: '02'
oa: 1
oa_version: Published Version
page: 947 - 990
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
grant_number: '338804'
name: Random matrices, universality and disordered quantum systems
publication: Communications in Mathematical Physics
publication_identifier:
issn:
- '00103616'
publication_status: published
publisher: Springer
publist_id: '6141'
pubrep_id: '722'
quality_controlled: '1'
status: public
title: Local law of addition of random matrices on optimal scale
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 349
year: '2017'
...
---
_id: '615'
abstract:
- lang: eng
text: We show that the Dyson Brownian Motion exhibits local universality after a
very short time assuming that local rigidity and level repulsion of the eigenvalues
hold. These conditions are verified, hence bulk spectral universality is proven,
for a large class of Wigner-like matrices, including deformed Wigner ensembles
and ensembles with non-stochastic variance matrices whose limiting densities differ
from Wigner's semicircle law.
author:
- first_name: László
full_name: Erdös, László
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
- first_name: Kevin
full_name: Schnelli, Kevin
id: 434AD0AE-F248-11E8-B48F-1D18A9856A87
last_name: Schnelli
citation:
ama: Erdös L, Schnelli K. Universality for random matrix flows with time dependent
density. *Annales de l’institut Henri Poincare (B) Probability and Statistics*.
2017;53(4):1606-1656. doi:10.1214/16-AIHP765
apa: Erdös, L., & Schnelli, K. (2017). Universality for random matrix flows
with time dependent density. *Annales de l’institut Henri Poincare (B) Probability
and Statistics*, *53*(4), 1606–1656. https://doi.org/10.1214/16-AIHP765
chicago: 'Erdös, László, and Kevin Schnelli. “Universality for Random Matrix Flows
with Time Dependent Density.” *Annales de l’institut Henri Poincare (B) Probability
and Statistics* 53, no. 4 (2017): 1606–56. https://doi.org/10.1214/16-AIHP765.'
ieee: L. Erdös and K. Schnelli, “Universality for random matrix flows with time
dependent density,” *Annales de l’institut Henri Poincare (B) Probability and
Statistics*, vol. 53, no. 4, pp. 1606–1656, 2017.
ista: Erdös L, Schnelli K. 2017. Universality for random matrix flows with time
dependent density. Annales de l’institut Henri Poincare (B) Probability and Statistics.
53(4), 1606–1656.
mla: Erdös, László, and Kevin Schnelli. “Universality for Random Matrix Flows with
Time Dependent Density.” *Annales de l’institut Henri Poincare (B) Probability
and Statistics*, vol. 53, no. 4, Institute of Mathematical Statistics, 2017,
pp. 1606–56, doi:10.1214/16-AIHP765.
short: L. Erdös, K. Schnelli, Annales de l’institut Henri Poincare (B) Probability
and Statistics 53 (2017) 1606–1656.
date_created: 2018-12-11T11:47:30Z
date_published: 2017-11-01T00:00:00Z
date_updated: 2019-08-02T12:39:11Z
day: '01'
department:
- _id: LaEr
doi: 10.1214/16-AIHP765
intvolume: ' 53'
issue: '4'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1504.00650
month: '11'
oa: 1
oa_version: Submitted Version
page: 1606 - 1656
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
grant_number: '338804'
name: Random matrices, universality and disordered quantum systems
publication: Annales de l'institut Henri Poincare (B) Probability and Statistics
publication_identifier:
issn:
- '02460203'
publication_status: published
publisher: Institute of Mathematical Statistics
publist_id: '7189'
quality_controlled: '1'
status: public
title: Universality for random matrix flows with time dependent density
type: journal_article
user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87
volume: 53
year: '2017'
...
---
_id: '1434'
abstract:
- lang: eng
text: We prove that the system of subordination equations, defining the free additive
convolution of two probability measures, is stable away from the edges of the
support and blow-up singularities by showing that the recent smoothness condition
of Kargin is always satisfied. As an application, we consider the local spectral
statistics of the random matrix ensemble A+UBU⁎A+UBU⁎, where U is a Haar distributed
random unitary or orthogonal matrix, and A and B are deterministic matrices.
In the bulk regime, we prove that the empirical spectral distribution of A+UBU⁎A+UBU⁎
concentrates around the free additive convolution of the spectral distributions
of A and B on scales down to N−2/3N−2/3.
author:
- first_name: Zhigang
full_name: Bao, Zhigang
id: 442E6A6C-F248-11E8-B48F-1D18A9856A87
last_name: Bao
orcid: 0000-0003-3036-1475
- first_name: László
full_name: Erdös, László
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
- first_name: Kevin
full_name: Schnelli, Kevin
id: 434AD0AE-F248-11E8-B48F-1D18A9856A87
last_name: Schnelli
citation:
ama: Bao Z, Erdös L, Schnelli K. Local stability of the free additive convolution.
*Journal of Functional Analysis*. 2016;271(3):672-719. doi:10.1016/j.jfa.2016.04.006
apa: Bao, Z., Erdös, L., & Schnelli, K. (2016). Local stability of the free
additive convolution. *Journal of Functional Analysis*, *271*(3), 672–719.
https://doi.org/10.1016/j.jfa.2016.04.006
chicago: 'Bao, Zhigang, László Erdös, and Kevin Schnelli. “Local Stability of the
Free Additive Convolution.” *Journal of Functional Analysis* 271, no. 3 (2016):
672–719. https://doi.org/10.1016/j.jfa.2016.04.006.'
ieee: Z. Bao, L. Erdös, and K. Schnelli, “Local stability of the free additive convolution,”
*Journal of Functional Analysis*, vol. 271, no. 3, pp. 672–719, 2016.
ista: Bao Z, Erdös L, Schnelli K. 2016. Local stability of the free additive convolution.
Journal of Functional Analysis. 271(3), 672–719.
mla: Bao, Zhigang, et al. “Local Stability of the Free Additive Convolution.” *Journal
of Functional Analysis*, vol. 271, no. 3, Academic Press, 2016, pp. 672–719,
doi:10.1016/j.jfa.2016.04.006.
short: Z. Bao, L. Erdös, K. Schnelli, Journal of Functional Analysis 271 (2016)
672–719.
date_created: 2018-12-11T11:52:00Z
date_published: 2016-08-01T00:00:00Z
date_updated: 2019-08-02T12:37:08Z
day: '01'
department:
- _id: LaEr
doi: 10.1016/j.jfa.2016.04.006
intvolume: ' 271'
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1508.05905
month: '08'
oa: 1
oa_version: Preprint
page: 672 - 719
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
grant_number: '338804'
name: Random matrices, universality and disordered quantum systems
publication: Journal of Functional Analysis
publication_status: published
publisher: Academic Press
publist_id: '5764'
quality_controlled: '1'
status: public
title: Local stability of the free additive convolution
type: journal_article
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 271
year: '2016'
...
---
_id: '1219'
abstract:
- lang: eng
text: We consider N×N random matrices of the form H = W + V where W is a real symmetric
or complex Hermitian Wigner matrix and V is 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 ofW and V are
typically of the same order. For a large class of diagonal matrices V , we show
that the local statistics in the bulk of the spectrum are universal in the limit
of large N.
acknowledgement: "J.C. was supported in part by National Research Foundation of Korea
Grant 2011-0013474 and TJ Park Junior Faculty Fellowship.\r\nK.S. was supported
by ERC Advanced Grant RANMAT, No. 338804, and the \"Fund for Math.\"\r\nB.S. was
supported by NSF GRFP Fellowship DGE-1144152.\r\nH.Y. was supported in part by NSF
Grant DMS-13-07444 and Simons investigator fellowship. We thank Paul Bourgade, László
Erd ̋os and Antti Knowles for helpful comments. We are grateful to the Taida Institute
for Mathematical\r\nSciences and National Taiwan Universality for their hospitality
during part of this\r\nresearch. We thank Thomas Spencer and the Institute for Advanced
Study for their\r\nhospitality during the academic year 2013–2014. "
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
- first_name: Ben
full_name: Stetler, Ben
last_name: Stetler
- first_name: Horngtzer
full_name: Yau, Horngtzer
last_name: Yau
citation:
ama: Lee J, Schnelli K, Stetler B, Yau H. Bulk universality for deformed wigner
matrices. *Annals of Probability*. 2016;44(3):2349-2425. doi:10.1214/15-AOP1023
apa: Lee, J., Schnelli, K., Stetler, B., & Yau, H. (2016). Bulk universality
for deformed wigner matrices. *Annals of Probability*, *44*(3), 2349–2425.
https://doi.org/10.1214/15-AOP1023
chicago: 'Lee, Jioon, Kevin Schnelli, Ben Stetler, and Horngtzer Yau. “Bulk Universality
for Deformed Wigner Matrices.” *Annals of Probability* 44, no. 3 (2016):
2349–2425. https://doi.org/10.1214/15-AOP1023.'
ieee: J. Lee, K. Schnelli, B. Stetler, and H. Yau, “Bulk universality for deformed
wigner matrices,” *Annals of Probability*, vol. 44, no. 3, pp. 2349–2425,
2016.
ista: Lee J, Schnelli K, Stetler B, Yau H. 2016. Bulk universality for deformed
wigner matrices. Annals of Probability. 44(3), 2349–2425.
mla: Lee, Jioon, et al. “Bulk Universality for Deformed Wigner Matrices.” *Annals
of Probability*, vol. 44, no. 3, Institute of Mathematical Statistics, 2016,
pp. 2349–425, doi:10.1214/15-AOP1023.
short: J. Lee, K. Schnelli, B. Stetler, H. Yau, Annals of Probability 44 (2016)
2349–2425.
date_created: 2018-12-11T11:50:47Z
date_published: 2016-01-01T00:00:00Z
date_updated: 2019-08-02T12:36:56Z
day: '01'
department:
- _id: LaEr
doi: 10.1214/15-AOP1023
intvolume: ' 44'
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1405.6634
month: '01'
oa: 1
oa_version: Preprint
page: 2349 - 2425
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
grant_number: '338804'
name: Random matrices, universality and disordered quantum systems
publication: Annals of Probability
publication_status: published
publisher: Institute of Mathematical Statistics
publist_id: '6115'
quality_controlled: '1'
status: public
title: Bulk universality for deformed wigner matrices
type: journal_article
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 44
year: '2016'
...
---
_id: '1157'
abstract:
- lang: eng
text: We consider sample covariance matrices of the form Q = ( σ1/2X)(σ1/2X)∗, where
the sample X is an M ×N random matrix whose entries are real independent random
variables with variance 1/N and whereσ is an M × M positive-definite deterministic
matrix. We analyze the asymptotic fluctuations of the largest rescaled eigenvalue
of Q when both M and N tend to infinity with N/M →d ϵ (0,∞). For a large class
of populations σ in the sub-critical regime, we show that the distribution of
the largest rescaled eigenvalue of Q is given by the type-1 Tracy-Widom distribution
under the additional assumptions that (1) either the entries of X are i.i.d. Gaussians
or (2) that σ is diagonal and that the entries of X have a sub-exponential decay.
acknowledgement: "We thank Horng-Tzer Yau for numerous discussions and remarks. We
are grateful to Ben Adlam, Jinho Baik, Zhigang Bao, Paul Bourgade, László Erd ̋os,
Iain Johnstone and Antti Knowles for comments. We are also grate-\r\nful to the
anonymous referee for carefully reading our manuscript and suggesting several improvements."
author:
- first_name: Ji
full_name: Lee, Ji
last_name: Lee
- first_name: Kevin
full_name: Schnelli, Kevin
id: 434AD0AE-F248-11E8-B48F-1D18A9856A87
last_name: Schnelli
citation:
ama: Lee J, Schnelli K. Tracy-widom distribution for the largest eigenvalue of real
sample covariance matrices with general population. *Annals of Applied Probability*.
2016;26(6):3786-3839. doi:10.1214/16-AAP1193
apa: Lee, J., & Schnelli, K. (2016). Tracy-widom distribution for the largest
eigenvalue of real sample covariance matrices with general population. *Annals
of Applied Probability*, *26*(6), 3786–3839. https://doi.org/10.1214/16-AAP1193
chicago: 'Lee, Ji, and Kevin Schnelli. “Tracy-Widom Distribution for the Largest
Eigenvalue of Real Sample Covariance Matrices with General Population.” *Annals
of Applied Probability* 26, no. 6 (2016): 3786–3839. https://doi.org/10.1214/16-AAP1193.'
ieee: J. Lee and K. Schnelli, “Tracy-widom distribution for the largest eigenvalue
of real sample covariance matrices with general population,” *Annals of Applied
Probability*, vol. 26, no. 6, pp. 3786–3839, 2016.
ista: Lee J, Schnelli K. 2016. Tracy-widom distribution for the largest eigenvalue
of real sample covariance matrices with general population. Annals of Applied
Probability. 26(6), 3786–3839.
mla: Lee, Ji, and Kevin Schnelli. “Tracy-Widom Distribution for the Largest Eigenvalue
of Real Sample Covariance Matrices with General Population.” *Annals of Applied
Probability*, vol. 26, no. 6, Institute of Mathematical Statistics, 2016, pp.
3786–839, doi:10.1214/16-AAP1193.
short: J. Lee, K. Schnelli, Annals of Applied Probability 26 (2016) 3786–3839.
date_created: 2018-12-11T11:50:27Z
date_published: 2016-12-15T00:00:00Z
date_updated: 2019-08-02T12:36:53Z
day: '15'
department:
- _id: LaEr
doi: 10.1214/16-AAP1193
intvolume: ' 26'
issue: '6'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1409.4979
month: '12'
oa: 1
oa_version: Preprint
page: 3786 - 3839
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
grant_number: '338804'
name: Random matrices, universality and disordered quantum systems
publication: Annals of Applied Probability
publication_status: published
publisher: Institute of Mathematical Statistics
publist_id: '6201'
quality_controlled: '1'
status: public
title: Tracy-widom distribution for the largest eigenvalue of real sample covariance
matrices with general population
type: journal_article
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 26
year: '2016'
...
---
_id: '1881'
abstract:
- lang: eng
text: 'We consider random matrices of the form H=W+λV, λ∈ℝ+, where W is a real symmetric
or complex Hermitian Wigner matrix of size N and V is a real bounded diagonal
random matrix of size N with i.i.d.\ entries that are independent of W. We assume
subexponential decay for the matrix entries of W and we choose λ∼1, so that the
eigenvalues of W and λV are typically of the same order. Further, we assume that
the density of the entries of V is supported on a single interval and is convex
near the edges of its support. In this paper we prove that there is λ+∈ℝ+ such
that the largest eigenvalues of H are in the limit of large N determined by the
order statistics of V for λ>λ+. In particular, the largest eigenvalue of H
has a Weibull distribution in the limit N→∞ if λ>λ+. Moreover, for N sufficiently
large, we show that the eigenvectors associated to the largest eigenvalues are
partially localized for λ>λ+, while they are completely delocalized for λ<λ+.
Similar results hold for the lowest eigenvalues. '
acknowledgement: "Most of the presented work was obtained while Kevin Schnelli was
staying at the IAS with the support of\r\nThe Fund For Math."
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
citation:
ama: Lee J, Schnelli K. Extremal eigenvalues and eigenvectors of deformed Wigner
matrices. *Probability Theory and Related Fields*. 2016;164(1-2):165-241.
doi:10.1007/s00440-014-0610-8
apa: Lee, J., & Schnelli, K. (2016). Extremal eigenvalues and eigenvectors of
deformed Wigner matrices. *Probability Theory and Related Fields*, *164*(1–2),
165–241. https://doi.org/10.1007/s00440-014-0610-8
chicago: 'Lee, Jioon, and Kevin Schnelli. “Extremal Eigenvalues and Eigenvectors
of Deformed Wigner Matrices.” *Probability Theory and Related Fields* 164,
no. 1–2 (2016): 165–241. https://doi.org/10.1007/s00440-014-0610-8.'
ieee: J. Lee and K. Schnelli, “Extremal eigenvalues and eigenvectors of deformed
Wigner matrices,” *Probability Theory and Related Fields*, vol. 164, no.
1–2, pp. 165–241, 2016.
ista: Lee J, Schnelli K. 2016. Extremal eigenvalues and eigenvectors of deformed
Wigner matrices. Probability Theory and Related Fields. 164(1–2), 165–241.
mla: Lee, Jioon, and Kevin Schnelli. “Extremal Eigenvalues and Eigenvectors of Deformed
Wigner Matrices.” *Probability Theory and Related Fields*, vol. 164, no.
1–2, Springer, 2016, pp. 165–241, doi:10.1007/s00440-014-0610-8.
short: J. Lee, K. Schnelli, Probability Theory and Related Fields 164 (2016) 165–241.
date_created: 2018-12-11T11:54:31Z
date_published: 2016-02-01T00:00:00Z
date_updated: 2019-08-02T12:37:29Z
day: '01'
department:
- _id: LaEr
doi: 10.1007/s00440-014-0610-8
intvolume: ' 164'
issue: 1-2
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1310.7057
month: '02'
oa: 1
oa_version: Preprint
page: 165 - 241
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
grant_number: '338804'
name: Random matrices, universality and disordered quantum systems
publication: Probability Theory and Related Fields
publication_status: published
publisher: Springer
publist_id: '5215'
quality_controlled: '1'
status: public
title: Extremal eigenvalues and eigenvectors of deformed Wigner matrices
type: journal_article
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 164
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
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*, *27*(8). https://doi.org/10.1142/S0129055X1550018X
chicago: Lee, Jioon, and Kevin Schnelli. “Edge Universality for Deformed Wigner
Matrices.” *Reviews in Mathematical Physics* 27, no. 8 (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, 2015.
ista: Lee J, Schnelli K. 2015. Edge universality for deformed Wigner matrices. Reviews
in Mathematical Physics. 27(8).
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: 2019-01-24T13:04:02Z
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'
status: public
title: Edge universality for deformed Wigner matrices
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 27
year: '2015'
...