--- _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 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' ...