---
_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'
...