---
_id: '2700'
alternative_title:
- Quantum Theory from Small to Large Scales
author:
- first_name: László
full_name: László Erdös
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
citation:
ama: 'Erdös L. Lecture notes on quantum Brownian motion. In: Vol 95. Oxford University
Press; 2012:3-98.'
apa: Erdös, L. (2012). Lecture notes on quantum Brownian motion (Vol. 95, pp. 3–98).
Presented at the Les Houches Summer School 2010, Oxford University Press.
chicago: Erdös, László. “Lecture Notes on Quantum Brownian Motion,” 95:3–98. Oxford
University Press, 2012.
ieee: L. Erdös, “Lecture notes on quantum Brownian motion,” presented at the Les
Houches Summer School 2010, 2012, vol. 95, pp. 3–98.
ista: Erdös L. 2012. Lecture notes on quantum Brownian motion. Les Houches Summer
School 2010, Quantum Theory from Small to Large Scales, vol. 95, 3–98.
mla: Erdös, László. Lecture Notes on Quantum Brownian Motion. Vol. 95, Oxford
University Press, 2012, pp. 3–98.
short: L. Erdös, in:, Oxford University Press, 2012, pp. 3–98.
conference:
name: Les Houches Summer School 2010
date_created: 2018-12-11T11:59:08Z
date_published: 2012-05-24T00:00:00Z
date_updated: 2021-01-12T06:59:08Z
day: '24'
extern: 1
intvolume: ' 95'
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1009.0843
month: '05'
oa: 1
page: 3 - 98
publication_status: published
publisher: Oxford University Press
publist_id: '4196'
quality_controlled: 0
status: public
title: Lecture notes on quantum Brownian motion
type: conference
volume: 95
year: '2012'
...
---
_id: '2715'
abstract:
- lang: eng
text: 'We consider Markov decision processes (MDPs) with specifications given as
Büchi (liveness) objectives. We consider the problem of computing the set of almost-sure
winning vertices from where the objective can be ensured with probability 1. We
study for the first time the average case complexity of the classical algorithm
for computing the set of almost-sure winning vertices for MDPs with Büchi objectives.
Our contributions are as follows: First, we show that for MDPs with constant out-degree
the expected number of iterations is at most logarithmic and the average case
running time is linear (as compared to the worst case linear number of iterations
and quadratic time complexity). Second, for the average case analysis over all
MDPs we show that the expected number of iterations is constant and the average
case running time is linear (again as compared to the worst case linear number
of iterations and quadratic time complexity). Finally we also show that given
that all MDPs are equally likely, the probability that the classical algorithm
requires more than constant number of iterations is exponentially small.'
alternative_title:
- LIPIcs
author:
- first_name: Krishnendu
full_name: Chatterjee, Krishnendu
id: 2E5DCA20-F248-11E8-B48F-1D18A9856A87
last_name: Chatterjee
orcid: 0000-0002-4561-241X
- first_name: Manas
full_name: Joglekar, Manas
last_name: Joglekar
- first_name: Nisarg
full_name: Shah, Nisarg
last_name: Shah
citation:
ama: 'Chatterjee K, Joglekar M, Shah N. Average case analysis of the classical algorithm
for Markov decision processes with Büchi objectives. In: Vol 18. Schloss Dagstuhl
- Leibniz-Zentrum für Informatik; 2012:461-473. doi:10.4230/LIPIcs.FSTTCS.2012.461'
apa: 'Chatterjee, K., Joglekar, M., & Shah, N. (2012). Average case analysis
of the classical algorithm for Markov decision processes with Büchi objectives
(Vol. 18, pp. 461–473). Presented at the FSTTCS: Foundations of Software Technology
and Theoretical Computer Science, Hyderabad, India: Schloss Dagstuhl - Leibniz-Zentrum
für Informatik. https://doi.org/10.4230/LIPIcs.FSTTCS.2012.461'
chicago: Chatterjee, Krishnendu, Manas Joglekar, and Nisarg Shah. “Average Case
Analysis of the Classical Algorithm for Markov Decision Processes with Büchi Objectives,”
18:461–73. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2012. https://doi.org/10.4230/LIPIcs.FSTTCS.2012.461.
ieee: 'K. Chatterjee, M. Joglekar, and N. Shah, “Average case analysis of the classical
algorithm for Markov decision processes with Büchi objectives,” presented at the
FSTTCS: Foundations of Software Technology and Theoretical Computer Science, Hyderabad,
India, 2012, vol. 18, pp. 461–473.'
ista: 'Chatterjee K, Joglekar M, Shah N. 2012. Average case analysis of the classical
algorithm for Markov decision processes with Büchi objectives. FSTTCS: Foundations
of Software Technology and Theoretical Computer Science, LIPIcs, vol. 18, 461–473.'
mla: Chatterjee, Krishnendu, et al. Average Case Analysis of the Classical Algorithm
for Markov Decision Processes with Büchi Objectives. Vol. 18, Schloss Dagstuhl
- Leibniz-Zentrum für Informatik, 2012, pp. 461–73, doi:10.4230/LIPIcs.FSTTCS.2012.461.
short: K. Chatterjee, M. Joglekar, N. Shah, in:, Schloss Dagstuhl - Leibniz-Zentrum
für Informatik, 2012, pp. 461–473.
conference:
end_date: 2012-12-17
location: Hyderabad, India
name: 'FSTTCS: Foundations of Software Technology and Theoretical Computer Science'
start_date: 2012-12-15
date_created: 2018-12-11T11:59:13Z
date_published: 2012-12-10T00:00:00Z
date_updated: 2023-02-23T10:06:04Z
day: '10'
ddc:
- '000'
department:
- _id: KrCh
doi: 10.4230/LIPIcs.FSTTCS.2012.461
ec_funded: 1
file:
- access_level: open_access
checksum: d4d644ed1a885dbfc4fa1ef4c5724dab
content_type: application/pdf
creator: system
date_created: 2018-12-12T10:13:53Z
date_updated: 2020-07-14T12:45:45Z
file_id: '5040'
file_name: IST-2016-525-v1+1_42_1_.pdf
file_size: 519040
relation: main_file
file_date_updated: 2020-07-14T12:45:45Z
has_accepted_license: '1'
intvolume: ' 18'
language:
- iso: eng
license: https://creativecommons.org/licenses/by-nc-nd/4.0/
month: '12'
oa: 1
oa_version: Published Version
page: 461 - 473
project:
- _id: 2584A770-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: P 23499-N23
name: Modern Graph Algorithmic Techniques in Formal Verification
- _id: 25863FF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: S11407
name: Game Theory
- _id: 2581B60A-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '279307'
name: 'Quantitative Graph Games: Theory and Applications'
- _id: 2587B514-B435-11E9-9278-68D0E5697425
name: Microsoft Research Faculty Fellowship
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
publist_id: '4180'
pubrep_id: '525'
quality_controlled: '1'
related_material:
record:
- id: '1598'
relation: later_version
status: public
scopus_import: 1
status: public
title: Average case analysis of the classical algorithm for Markov decision processes
with Büchi objectives
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: conference
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 18
year: '2012'
...
---
_id: '10904'
abstract:
- lang: eng
text: Multi-dimensional mean-payoff and energy games provide the mathematical foundation
for the quantitative study of reactive systems, and play a central role in the
emerging quantitative theory of verification and synthesis. In this work, we study
the strategy synthesis problem for games with such multi-dimensional objectives
along with a parity condition, a canonical way to express ω-regular conditions.
While in general, the winning strategies in such games may require infinite memory,
for synthesis the most relevant problem is the construction of a finite-memory
winning strategy (if one exists). Our main contributions are as follows. First,
we show a tight exponential bound (matching upper and lower bounds) on the memory
required for finite-memory winning strategies in both multi-dimensional mean-payoff
and energy games along with parity objectives. This significantly improves the
triple exponential upper bound for multi energy games (without parity) that could
be derived from results in literature for games on VASS (vector addition systems
with states). Second, we present an optimal symbolic and incremental algorithm
to compute a finite-memory winning strategy (if one exists) in such games. Finally,
we give a complete characterization of when finite memory of strategies can be
traded off for randomness. In particular, we show that for one-dimension mean-payoff
parity games, randomized memoryless strategies are as powerful as their pure finite-memory
counterparts.
acknowledgement: 'Author supported by Austrian Science Fund (FWF) Grant No P 23499-N23,
FWF NFN Grant No S11407 (RiSE), ERC Start Grant (279307: Graph Games), Microsoft
faculty fellowship.'
alternative_title:
- LNCS
article_processing_charge: No
author:
- first_name: Krishnendu
full_name: Chatterjee, Krishnendu
id: 2E5DCA20-F248-11E8-B48F-1D18A9856A87
last_name: Chatterjee
orcid: 0000-0002-4561-241X
- first_name: Mickael
full_name: Randour, Mickael
last_name: Randour
- first_name: Jean-François
full_name: Raskin, Jean-François
last_name: Raskin
citation:
ama: 'Chatterjee K, Randour M, Raskin J-F. Strategy synthesis for multi-dimensional
quantitative objectives. In: Koutny M, Ulidowski I, eds. CONCUR 2012 - Concurrency
Theory. Vol 7454. Berlin, Heidelberg: Springer; 2012:115-131. doi:10.1007/978-3-642-32940-1_10'
apa: 'Chatterjee, K., Randour, M., & Raskin, J.-F. (2012). Strategy synthesis
for multi-dimensional quantitative objectives. In M. Koutny & I. Ulidowski
(Eds.), CONCUR 2012 - Concurrency Theory (Vol. 7454, pp. 115–131). Berlin,
Heidelberg: Springer. https://doi.org/10.1007/978-3-642-32940-1_10'
chicago: 'Chatterjee, Krishnendu, Mickael Randour, and Jean-François Raskin. “Strategy
Synthesis for Multi-Dimensional Quantitative Objectives.” In CONCUR 2012 -
Concurrency Theory, edited by Maciej Koutny and Irek Ulidowski, 7454:115–31.
Berlin, Heidelberg: Springer, 2012. https://doi.org/10.1007/978-3-642-32940-1_10.'
ieee: K. Chatterjee, M. Randour, and J.-F. Raskin, “Strategy synthesis for multi-dimensional
quantitative objectives,” in CONCUR 2012 - Concurrency Theory, Newcastle
upon Tyne, United Kingdom, 2012, vol. 7454, pp. 115–131.
ista: 'Chatterjee K, Randour M, Raskin J-F. 2012. Strategy synthesis for multi-dimensional
quantitative objectives. CONCUR 2012 - Concurrency Theory. CONCUR: Conference
on Concurrency Theory, LNCS, vol. 7454, 115–131.'
mla: Chatterjee, Krishnendu, et al. “Strategy Synthesis for Multi-Dimensional Quantitative
Objectives.” CONCUR 2012 - Concurrency Theory, edited by Maciej Koutny
and Irek Ulidowski, vol. 7454, Springer, 2012, pp. 115–31, doi:10.1007/978-3-642-32940-1_10.
short: K. Chatterjee, M. Randour, J.-F. Raskin, in:, M. Koutny, I. Ulidowski (Eds.),
CONCUR 2012 - Concurrency Theory, Springer, Berlin, Heidelberg, 2012, pp. 115–131.
conference:
end_date: 2012-09-07
location: Newcastle upon Tyne, United Kingdom
name: 'CONCUR: Conference on Concurrency Theory'
start_date: 2012-09-04
date_created: 2022-03-21T08:00:21Z
date_published: 2012-09-15T00:00:00Z
date_updated: 2023-02-23T10:55:06Z
day: '15'
department:
- _id: KrCh
doi: 10.1007/978-3-642-32940-1_10
ec_funded: 1
editor:
- first_name: Maciej
full_name: Koutny, Maciej
last_name: Koutny
- first_name: Irek
full_name: Ulidowski, Irek
last_name: Ulidowski
external_id:
arxiv:
- '1201.5073'
intvolume: ' 7454'
language:
- iso: eng
month: '09'
oa_version: Preprint
page: 115-131
place: Berlin, Heidelberg
project:
- _id: 2584A770-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: P 23499-N23
name: Modern Graph Algorithmic Techniques in Formal Verification
- _id: 25863FF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: S11407
name: Game Theory
- _id: 2581B60A-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '279307'
name: 'Quantitative Graph Games: Theory and Applications'
- _id: 2587B514-B435-11E9-9278-68D0E5697425
name: Microsoft Research Faculty Fellowship
publication: CONCUR 2012 - Concurrency Theory
publication_identifier:
eisbn:
- '9783642329401'
isbn:
- '9783642329395'
issn:
- 0302-9743
- 1611-3349
publication_status: published
publisher: Springer
quality_controlled: '1'
related_material:
record:
- id: '2716'
relation: later_version
status: public
scopus_import: '1'
status: public
title: Strategy synthesis for multi-dimensional quantitative objectives
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 7454
year: '2012'
...
---
_id: '2770'
abstract:
- lang: eng
text: 'Consider N×N Hermitian or symmetric random matrices H with independent entries,
where the distribution of the (i,j) matrix element is given by the probability
measure vij with zero expectation and with variance σ ιj 2. We assume that the
variances satisfy the normalization condition Σiσij2=1 for all j and that there
is a positive constant c such that c≤Nσ ιj 2 ιc -1. We further assume that the
probability distributions νij have a uniform subexponential decay. We prove that
the Stieltjes transform of the empirical eigenvalue distribution of H is given
by the Wigner semicircle law uniformly up to the edges of the spectrum with an
error of order (Nη) -1 where η is the imaginary part of the spectral parameter
in the Stieltjes transform. There are three corollaries to this strong local semicircle
law: (1) Rigidity of eigenvalues: If γj=γj,N denotes the classical location of
the j-th eigenvalue under the semicircle law ordered in increasing order, then
the j-th eigenvalue λj is close to γj in the sense that for some positive constants
C, c P{double-struck}(∃j:|λ j-γ j|≥(logN) CloglogN[min(j,N-j+1)] -1/3N -2/3)≤
C exp[-(logN) cloglogN] for N large enough. (2) The proof of Dyson''s conjecture
(Dyson, 1962 [15]) which states that the time scale of the Dyson Brownian motion
to reach local equilibrium is of order N -1 up to logarithmic corrections. (3)
The edge universality holds in the sense that the probability distributions of
the largest (and the smallest) eigenvalues of two generalized Wigner ensembles
are the same in the large N limit provided that the second moments of the two
ensembles are identical.'
author:
- first_name: László
full_name: László Erdös
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
- first_name: Horng
full_name: Yau, Horng-Tzer
last_name: Yau
- first_name: Jun
full_name: Yin, Jun
last_name: Yin
citation:
ama: Erdös L, Yau H, Yin J. Rigidity of eigenvalues of generalized Wigner matrices.
Advances in Mathematics. 2012;229(3):1435-1515. doi:10.1016/j.aim.2011.12.010
apa: Erdös, L., Yau, H., & Yin, J. (2012). Rigidity of eigenvalues of generalized
Wigner matrices. Advances in Mathematics. Academic Press. https://doi.org/10.1016/j.aim.2011.12.010
chicago: Erdös, László, Horng Yau, and Jun Yin. “Rigidity of Eigenvalues of Generalized
Wigner Matrices.” Advances in Mathematics. Academic Press, 2012. https://doi.org/10.1016/j.aim.2011.12.010.
ieee: L. Erdös, H. Yau, and J. Yin, “Rigidity of eigenvalues of generalized Wigner
matrices,” Advances in Mathematics, vol. 229, no. 3. Academic Press, pp.
1435–1515, 2012.
ista: Erdös L, Yau H, Yin J. 2012. Rigidity of eigenvalues of generalized Wigner
matrices. Advances in Mathematics. 229(3), 1435–1515.
mla: Erdös, László, et al. “Rigidity of Eigenvalues of Generalized Wigner Matrices.”
Advances in Mathematics, vol. 229, no. 3, Academic Press, 2012, pp. 1435–515,
doi:10.1016/j.aim.2011.12.010.
short: L. Erdös, H. Yau, J. Yin, Advances in Mathematics 229 (2012) 1435–1515.
date_created: 2018-12-11T11:59:30Z
date_published: 2012-02-15T00:00:00Z
date_updated: 2021-01-12T06:59:35Z
day: '15'
doi: 10.1016/j.aim.2011.12.010
extern: 1
intvolume: ' 229'
issue: '3'
month: '02'
page: 1435 - 1515
publication: Advances in Mathematics
publication_status: published
publisher: Academic Press
publist_id: '4120'
quality_controlled: 0
status: public
title: Rigidity of eigenvalues of generalized Wigner matrices
type: journal_article
volume: 229
year: '2012'
...
---
_id: '2769'
abstract:
- lang: eng
text: We present a generalization of the method of the local relaxation flow to
establish the universality of local spectral statistics of a broad class of large
random matrices. We show that the local distribution of the eigenvalues coincides
with the local statistics of the corresponding Gaussian ensemble provided the
distribution of the individual matrix element is smooth and the eigenvalues {X
J} N j=1 are close to their classical location {y j} N j=1 determined by the limiting
density of eigenvalues. Under the scaling where the typical distance between neighboring
eigenvalues is of order 1/N, the necessary apriori estimate on the location of
eigenvalues requires only to know that E|x j - γ j| 2 ≤ N-1-ε on average. This
information can be obtained by well established methods for various matrix ensembles.
We demonstrate the method by proving local spectral universality for sample covariance
matrices.
author:
- first_name: László
full_name: László Erdös
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
- first_name: Benjamin
full_name: Schlein, Benjamin
last_name: Schlein
- first_name: Horng
full_name: Yau, Horng-Tzer
last_name: Yau
- first_name: Jun
full_name: Yin, Jun
last_name: Yin
citation:
ama: Erdös L, Schlein B, Yau H, Yin J. The local relaxation flow approach to universality
of the local statistics for random matrices. Annales de l’institut Henri Poincare
(B) Probability and Statistics. 2012;48(1):1-46. doi:10.1214/10-AIHP388
apa: Erdös, L., Schlein, B., Yau, H., & Yin, J. (2012). The local relaxation
flow approach to universality of the local statistics for random matrices. Annales
de l’institut Henri Poincare (B) Probability and Statistics. Institute of
Mathematical Statistics. https://doi.org/10.1214/10-AIHP388
chicago: Erdös, László, Benjamin Schlein, Horng Yau, and Jun Yin. “The Local Relaxation
Flow Approach to Universality of the Local Statistics for Random Matrices.” Annales
de l’institut Henri Poincare (B) Probability and Statistics. Institute of
Mathematical Statistics, 2012. https://doi.org/10.1214/10-AIHP388.
ieee: L. Erdös, B. Schlein, H. Yau, and J. Yin, “The local relaxation flow approach
to universality of the local statistics for random matrices,” Annales de l’institut
Henri Poincare (B) Probability and Statistics, vol. 48, no. 1. Institute of
Mathematical Statistics, pp. 1–46, 2012.
ista: Erdös L, Schlein B, Yau H, Yin J. 2012. The local relaxation flow approach
to universality of the local statistics for random matrices. Annales de l’institut
Henri Poincare (B) Probability and Statistics. 48(1), 1–46.
mla: Erdös, László, et al. “The Local Relaxation Flow Approach to Universality of
the Local Statistics for Random Matrices.” Annales de l’institut Henri Poincare
(B) Probability and Statistics, vol. 48, no. 1, Institute of Mathematical
Statistics, 2012, pp. 1–46, doi:10.1214/10-AIHP388.
short: L. Erdös, B. Schlein, H. Yau, J. Yin, Annales de l’institut Henri Poincare
(B) Probability and Statistics 48 (2012) 1–46.
date_created: 2018-12-11T11:59:30Z
date_published: 2012-02-01T00:00:00Z
date_updated: 2021-01-12T06:59:34Z
day: '01'
doi: 10.1214/10-AIHP388
extern: 1
intvolume: ' 48'
issue: '1'
month: '02'
page: 1 - 46
publication: Annales de l'institut Henri Poincare (B) Probability and Statistics
publication_status: published
publisher: Institute of Mathematical Statistics
publist_id: '4121'
quality_controlled: 0
status: public
title: The local relaxation flow approach to universality of the local statistics
for random matrices
type: journal_article
volume: 48
year: '2012'
...
---
_id: '2767'
abstract:
- lang: eng
text: 'Consider N × N Hermitian or symmetric random matrices H where the distribution
of the (i, j) matrix element is given by a probability measure ν ij with a subexponential
decay. Let σ ij 2 be the variance for the probability measure ν ij with the normalization
property that Σ iσ i,j 2 = 1 for all j. Under essentially the only condition that
c ≤ N σ ij 2 ≤ c -1 for some constant c > 0, we prove that, in the limit N
→ ∞, the eigenvalue spacing statistics of H in the bulk of the spectrum coincide
with those of the Gaussian unitary or orthogonal ensemble (GUE or GOE). We also
show that for band matrices with bandwidth M the local semicircle law holds to
the energy scale M -1. '
author:
- first_name: László
full_name: László Erdös
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
- first_name: Horng
full_name: Yau, Horng-Tzer
last_name: Yau
- first_name: Jun
full_name: Yin, Jun
last_name: Yin
citation:
ama: Erdös L, Yau H, Yin J. Bulk universality for generalized Wigner matrices. Probability
Theory and Related Fields. 2012;154(1-2):341-407. doi:10.1007/s00440-011-0390-3
apa: Erdös, L., Yau, H., & Yin, J. (2012). Bulk universality for generalized
Wigner matrices. Probability Theory and Related Fields. Springer. https://doi.org/10.1007/s00440-011-0390-3
chicago: Erdös, László, Horng Yau, and Jun Yin. “Bulk Universality for Generalized
Wigner Matrices.” Probability Theory and Related Fields. Springer, 2012.
https://doi.org/10.1007/s00440-011-0390-3.
ieee: L. Erdös, H. Yau, and J. Yin, “Bulk universality for generalized Wigner matrices,”
Probability Theory and Related Fields, vol. 154, no. 1–2. Springer, pp.
341–407, 2012.
ista: Erdös L, Yau H, Yin J. 2012. Bulk universality for generalized Wigner matrices.
Probability Theory and Related Fields. 154(1–2), 341–407.
mla: Erdös, László, et al. “Bulk Universality for Generalized Wigner Matrices.”
Probability Theory and Related Fields, vol. 154, no. 1–2, Springer, 2012,
pp. 341–407, doi:10.1007/s00440-011-0390-3.
short: L. Erdös, H. Yau, J. Yin, Probability Theory and Related Fields 154 (2012)
341–407.
date_created: 2018-12-11T11:59:29Z
date_published: 2012-10-01T00:00:00Z
date_updated: 2021-01-12T06:59:33Z
day: '01'
doi: 10.1007/s00440-011-0390-3
extern: 1
intvolume: ' 154'
issue: 1-2
month: '10'
page: 341 - 407
publication: Probability Theory and Related Fields
publication_status: published
publisher: Springer
publist_id: '4123'
quality_controlled: 0
status: public
title: Bulk universality for generalized Wigner matrices
type: journal_article
volume: 154
year: '2012'
...
---
_id: '2768'
abstract:
- lang: eng
text: We consider a two dimensional magnetic Schrödinger operator with a spatially
stationary random magnetic field. We assume that the magnetic field has a positive
lower bound and that it has Fourier modes on arbitrarily short scales. We prove
the Wegner estimate at arbitrary energy, i. e. we show that the averaged density
of states is finite throughout the whole spectrum. We also prove Anderson localization
at the bottom of the spectrum.
author:
- first_name: László
full_name: László Erdös
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
- first_name: David
full_name: Hasler, David G
last_name: Hasler
citation:
ama: Erdös L, Hasler D. Wegner estimate and Anderson localization for random magnetic
fields. Communications in Mathematical Physics. 2012;309(2):507-542. doi:10.1007/s00220-011-1373-z
apa: Erdös, L., & Hasler, D. (2012). Wegner estimate and Anderson localization
for random magnetic fields. Communications in Mathematical Physics. Springer.
https://doi.org/10.1007/s00220-011-1373-z
chicago: Erdös, László, and David Hasler. “Wegner Estimate and Anderson Localization
for Random Magnetic Fields.” Communications in Mathematical Physics. Springer,
2012. https://doi.org/10.1007/s00220-011-1373-z.
ieee: L. Erdös and D. Hasler, “Wegner estimate and Anderson localization for random
magnetic fields,” Communications in Mathematical Physics, vol. 309, no.
2. Springer, pp. 507–542, 2012.
ista: Erdös L, Hasler D. 2012. Wegner estimate and Anderson localization for random
magnetic fields. Communications in Mathematical Physics. 309(2), 507–542.
mla: Erdös, László, and David Hasler. “Wegner Estimate and Anderson Localization
for Random Magnetic Fields.” Communications in Mathematical Physics, vol.
309, no. 2, Springer, 2012, pp. 507–42, doi:10.1007/s00220-011-1373-z.
short: L. Erdös, D. Hasler, Communications in Mathematical Physics 309 (2012) 507–542.
date_created: 2018-12-11T11:59:30Z
date_published: 2012-01-01T00:00:00Z
date_updated: 2021-01-12T06:59:34Z
day: '01'
doi: 10.1007/s00220-011-1373-z
extern: 1
intvolume: ' 309'
issue: '2'
month: '01'
page: 507 - 542
publication: Communications in Mathematical Physics
publication_status: published
publisher: Springer
publist_id: '4122'
quality_controlled: 0
status: public
title: Wegner estimate and Anderson localization for random magnetic fields
type: journal_article
volume: 309
year: '2012'
...
---
_id: '2775'
abstract:
- lang: eng
text: The Wigner-Dyson-Gaudin-Mehta conjecture asserts that the local eigenvalue
statistics of large random matrices exhibit universal behavior depending only
on the symmetry class of the matrix ensemble. For invariant matrix models, the
eigenvalue distributions are given by a log-gas with potential V and inverse temperature
β = 1, 2, 4, corresponding to the orthogonal, unitary and symplectic ensembles.
For β ∉ {1, 2, 4}, there is no natural random matrix ensemble behind this model,
but the statistical physics interpretation of the log-gas is still valid for all
β > 0. The universality conjecture for invariant ensembles asserts that the
local eigenvalue statistics are independent of V. In this article, we review our
recent solution to the universality conjecture for both invariant and non-invariant
ensembles. We will also demonstrate that the local ergodicity of the Dyson Brownian
motion is the intrinsic mechanism behind the universality. Furthermore, we review
the solution of Dyson's conjecture on the local relaxation time of the Dyson Brownian
motion. Related questions such as delocalization of eigenvectors and local version
of Wigner's semicircle law will also be discussed.
author:
- first_name: László
full_name: László Erdös
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
- first_name: Horng
full_name: Yau, Horng-Tzer
last_name: Yau
citation:
ama: Erdös L, Yau H. Universality of local spectral statistics of random matrices.
Bulletin of the American Mathematical Society. 2012;49(3):377-414. doi:10.1090/S0273-0979-2012-01372-1
apa: Erdös, L., & Yau, H. (2012). Universality of local spectral statistics
of random matrices. Bulletin of the American Mathematical Society. American
Mathematical Society. https://doi.org/10.1090/S0273-0979-2012-01372-1
chicago: Erdös, László, and Horng Yau. “Universality of Local Spectral Statistics
of Random Matrices.” Bulletin of the American Mathematical Society. American
Mathematical Society, 2012. https://doi.org/10.1090/S0273-0979-2012-01372-1.
ieee: L. Erdös and H. Yau, “Universality of local spectral statistics of random
matrices,” Bulletin of the American Mathematical Society, vol. 49, no.
3. American Mathematical Society, pp. 377–414, 2012.
ista: Erdös L, Yau H. 2012. Universality of local spectral statistics of random
matrices. Bulletin of the American Mathematical Society. 49(3), 377–414.
mla: Erdös, László, and Horng Yau. “Universality of Local Spectral Statistics of
Random Matrices.” Bulletin of the American Mathematical Society, vol. 49,
no. 3, American Mathematical Society, 2012, pp. 377–414, doi:10.1090/S0273-0979-2012-01372-1.
short: L. Erdös, H. Yau, Bulletin of the American Mathematical Society 49 (2012)
377–414.
date_created: 2018-12-11T11:59:32Z
date_published: 2012-01-30T00:00:00Z
date_updated: 2021-01-12T06:59:36Z
day: '30'
doi: 10.1090/S0273-0979-2012-01372-1
extern: 1
intvolume: ' 49'
issue: '3'
month: '01'
page: 377 - 414
publication: Bulletin of the American Mathematical Society
publication_status: published
publisher: American Mathematical Society
publist_id: '4115'
quality_controlled: 0
status: public
title: Universality of local spectral statistics of random matrices
type: journal_article
volume: 49
year: '2012'
...
---
_id: '2777'
abstract:
- lang: eng
text: We consider a large neutral molecule with total nuclear charge Z in a model
with self-generated classical magnetic field and where the kinetic energy of the
electrons is treated relativistically. To ensure stability, we assume that Zα
< 2/π, where α denotes the fine structure constant. We are interested in the
ground state energy in the simultaneous limit Z → ∞, α → 0 such that κ = Zα is
fixed. The leading term in the energy asymptotics is independent of κ, it is given
by the Thomas-Fermi energy of order Z7/3 and it is unchanged by including the
self-generated magnetic field. We prove the first correction term to this energy,
the so-called Scott correction of the form S(αZ)Z2. The current paper extends
the result of Solovej et al. [Commun. Pure Appl. Math.LXIII, 39-118 (2010)] on
the Scott correction for relativistic molecules to include a self-generated magnetic
field. Furthermore, we show that the corresponding Scott correction function S,
first identified by Solovej et al. [Commun. Pure Appl. Math.LXIII, 39-118 (2010)],
is unchanged by including a magnetic field. We also prove new Lieb-Thirring inequalities
for the relativistic kinetic energy with magnetic fields.
author:
- first_name: László
full_name: László Erdös
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 P
last_name: Solovej
citation:
ama: Erdös L, Fournais S, Solovej J. Relativistic Scott correction in self-generated
magnetic fields. Journal of Mathematical Physics. 2012;53(9). doi:10.1063/1.3697417
apa: Erdös, L., Fournais, S., & Solovej, J. (2012). Relativistic Scott correction
in self-generated magnetic fields. Journal of Mathematical Physics. American
Institute of Physics. https://doi.org/10.1063/1.3697417
chicago: Erdös, László, Søren Fournais, and Jan Solovej. “Relativistic Scott Correction
in Self-Generated Magnetic Fields.” Journal of Mathematical Physics. American
Institute of Physics, 2012. https://doi.org/10.1063/1.3697417.
ieee: L. Erdös, S. Fournais, and J. Solovej, “Relativistic Scott correction in self-generated
magnetic fields,” Journal of Mathematical Physics, vol. 53, no. 9. American
Institute of Physics, 2012.
ista: Erdös L, Fournais S, Solovej J. 2012. Relativistic Scott correction in self-generated
magnetic fields. Journal of Mathematical Physics. 53(9).
mla: Erdös, László, et al. “Relativistic Scott Correction in Self-Generated Magnetic
Fields.” Journal of Mathematical Physics, vol. 53, no. 9, American Institute
of Physics, 2012, doi:10.1063/1.3697417.
short: L. Erdös, S. Fournais, J. Solovej, Journal of Mathematical Physics 53 (2012).
date_created: 2018-12-11T11:59:32Z
date_published: 2012-09-28T00:00:00Z
date_updated: 2021-01-12T06:59:37Z
day: '28'
doi: 10.1063/1.3697417
extern: 1
intvolume: ' 53'
issue: '9'
month: '09'
publication: Journal of Mathematical Physics
publication_status: published
publisher: American Institute of Physics
publist_id: '4113'
quality_controlled: 0
status: public
title: Relativistic Scott correction in self-generated magnetic fields
type: journal_article
volume: 53
year: '2012'
...
---
_id: '2772'
abstract:
- lang: eng
text: We consider the semiclassical asymptotics of the sum of negative eigenvalues
of the three-dimensional Pauli operator with an external potential and a self-generated
magnetic field B. We also add the field energy β ∫ B 2 and we minimize over all
magnetic fields. The parameter β effectively determines the strength of the field.
We consider the weak field regime with βh 2 ≥ const > 0, where h is the semiclassical
parameter. For smooth potentials we prove that the semiclassical asymptotics of
the total energy is given by the non-magnetic Weyl term to leading order with
an error bound that is smaller by a factor h 1+e{open}, i. e. the subleading term
vanishes. However for potentials with a Coulomb singularity, the subleading term
does not vanish due to the non-semiclassical effect of the singularity. Combined
with a multiscale technique, this refined estimate is used in the companion paper
(Erdo{double acute}s et al. in Scott correction for large molecules with a self-generated
magnetic field, Preprint, 2011) to prove the second order Scott correction to
the ground state energy of large atoms and molecules.
author:
- first_name: László
full_name: László Erdös
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 P
last_name: Solovej
citation:
ama: Erdös L, Fournais S, Solovej J. Second order semiclassics with self generated
magnetic fields. Annales Henri Poincare. 2012;13(4):671-730. doi:10.1007/s00023-011-0150-z
apa: Erdös, L., Fournais, S., & Solovej, J. (2012). Second order semiclassics
with self generated magnetic fields. Annales Henri Poincare. Birkhäuser.
https://doi.org/10.1007/s00023-011-0150-z
chicago: Erdös, László, Søren Fournais, and Jan Solovej. “Second Order Semiclassics
with Self Generated Magnetic Fields.” Annales Henri Poincare. Birkhäuser,
2012. https://doi.org/10.1007/s00023-011-0150-z.
ieee: L. Erdös, S. Fournais, and J. Solovej, “Second order semiclassics with self
generated magnetic fields,” Annales Henri Poincare, vol. 13, no. 4. Birkhäuser,
pp. 671–730, 2012.
ista: Erdös L, Fournais S, Solovej J. 2012. Second order semiclassics with self
generated magnetic fields. Annales Henri Poincare. 13(4), 671–730.
mla: Erdös, László, et al. “Second Order Semiclassics with Self Generated Magnetic
Fields.” Annales Henri Poincare, vol. 13, no. 4, Birkhäuser, 2012, pp.
671–730, doi:10.1007/s00023-011-0150-z.
short: L. Erdös, S. Fournais, J. Solovej, Annales Henri Poincare 13 (2012) 671–730.
date_created: 2018-12-11T11:59:31Z
date_published: 2012-05-01T00:00:00Z
date_updated: 2021-01-12T06:59:36Z
day: '01'
doi: 10.1007/s00023-011-0150-z
extern: 1
intvolume: ' 13'
issue: '4'
month: '05'
page: 671 - 730
publication: Annales Henri Poincare
publication_status: published
publisher: Birkhäuser
publist_id: '4118'
quality_controlled: 0
status: public
title: Second order semiclassics with self generated magnetic fields
type: journal_article
volume: 13
year: '2012'
...