[{"date_updated":"2023-02-23T10:55:06Z","department":[{"_id":"KrCh"}],"_id":"10904","status":"public","conference":{"name":"CONCUR: Conference on Concurrency Theory","location":"Newcastle upon Tyne, United Kingdom","end_date":"2012-09-07","start_date":"2012-09-04"},"type":"conference","language":[{"iso":"eng"}],"publication_status":"published","publication_identifier":{"issn":["0302-9743","1611-3349"],"isbn":["9783642329395"],"eisbn":["9783642329401"]},"ec_funded":1,"volume":7454,"related_material":{"record":[{"relation":"later_version","id":"2716","status":"public"}]},"oa_version":"Preprint","abstract":[{"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.","lang":"eng"}],"intvolume":" 7454","month":"09","place":"Berlin, Heidelberg","scopus_import":"1","alternative_title":["LNCS"],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","citation":{"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.","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.","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.","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.","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","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","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."},"title":"Strategy synthesis for multi-dimensional quantitative objectives","editor":[{"first_name":"Maciej","last_name":"Koutny","full_name":"Koutny, Maciej"},{"first_name":"Irek","full_name":"Ulidowski, Irek","last_name":"Ulidowski"}],"article_processing_charge":"No","external_id":{"arxiv":["1201.5073"]},"author":[{"id":"2E5DCA20-F248-11E8-B48F-1D18A9856A87","first_name":"Krishnendu","orcid":"0000-0002-4561-241X","full_name":"Chatterjee, Krishnendu","last_name":"Chatterjee"},{"full_name":"Randour, Mickael","last_name":"Randour","first_name":"Mickael"},{"full_name":"Raskin, Jean-François","last_name":"Raskin","first_name":"Jean-François"}],"project":[{"call_identifier":"FWF","_id":"2584A770-B435-11E9-9278-68D0E5697425","grant_number":"P 23499-N23","name":"Modern Graph Algorithmic Techniques in Formal Verification"},{"grant_number":"S11407","name":"Game Theory","call_identifier":"FWF","_id":"25863FF4-B435-11E9-9278-68D0E5697425"},{"call_identifier":"FP7","_id":"2581B60A-B435-11E9-9278-68D0E5697425","name":"Quantitative Graph Games: Theory and Applications","grant_number":"279307"},{"_id":"2587B514-B435-11E9-9278-68D0E5697425","name":"Microsoft Research Faculty Fellowship"}],"publication":"CONCUR 2012 - Concurrency Theory","day":"15","year":"2012","date_created":"2022-03-21T08:00:21Z","doi":"10.1007/978-3-642-32940-1_10","date_published":"2012-09-15T00:00:00Z","page":"115-131","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.","quality_controlled":"1","publisher":"Springer"},{"page":"1435 - 1515","date_published":"2012-02-15T00:00:00Z","volume":229,"issue":"3","doi":"10.1016/j.aim.2011.12.010","date_created":"2018-12-11T11:59:30Z","year":"2012","publication_status":"published","day":"15","publication":"Advances in Mathematics","publisher":"Academic Press","quality_controlled":0,"month":"02","intvolume":" 229","abstract":[{"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.","lang":"eng"}],"author":[{"orcid":"0000-0001-5366-9603","full_name":"László Erdös","last_name":"Erdös","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László"},{"first_name":"Horng","last_name":"Yau","full_name":"Yau, Horng-Tzer"},{"first_name":"Jun","last_name":"Yin","full_name":"Yin, Jun"}],"publist_id":"4120","title":"Rigidity of eigenvalues of generalized Wigner matrices","citation":{"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.","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.","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","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","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.","ista":"Erdös L, Yau H, Yin J. 2012. Rigidity of eigenvalues of generalized Wigner matrices. Advances in Mathematics. 229(3), 1435–1515."},"date_updated":"2021-01-12T06:59:35Z","extern":1,"type":"journal_article","status":"public","_id":"2770"},{"extern":1,"date_updated":"2021-01-12T06:59:34Z","citation":{"short":"L. Erdös, B. Schlein, H. Yau, J. Yin, Annales de l’institut Henri Poincare (B) Probability and Statistics 48 (2012) 1–46.","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.","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","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.","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.","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."},"title":"The local relaxation flow approach to universality of the local statistics for random matrices","publist_id":"4121","author":[{"last_name":"Erdös","orcid":"0000-0001-5366-9603","full_name":"László Erdös","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László"},{"first_name":"Benjamin","last_name":"Schlein","full_name":"Schlein, Benjamin"},{"first_name":"Horng","last_name":"Yau","full_name":"Yau, Horng-Tzer"},{"first_name":"Jun","full_name":"Yin, Jun","last_name":"Yin"}],"_id":"2769","status":"public","type":"journal_article","day":"01","publication":"Annales de l'institut Henri Poincare (B) Probability and Statistics","publication_status":"published","year":"2012","date_published":"2012-02-01T00:00:00Z","issue":"1","doi":"10.1214/10-AIHP388","volume":48,"date_created":"2018-12-11T11:59:30Z","page":"1 - 46","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."}],"month":"02","intvolume":" 48","publisher":"Institute of Mathematical Statistics","quality_controlled":0},{"author":[{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","last_name":"Erdös","full_name":"László Erdös","orcid":"0000-0001-5366-9603"},{"last_name":"Yau","full_name":"Yau, Horng-Tzer","first_name":"Horng"},{"first_name":"Jun","full_name":"Yin, Jun","last_name":"Yin"}],"publist_id":"4123","title":"Bulk universality for generalized Wigner matrices","date_updated":"2021-01-12T06:59:33Z","citation":{"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","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","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.","short":"L. Erdös, H. Yau, J. Yin, Probability Theory and Related Fields 154 (2012) 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.","ista":"Erdös L, Yau H, Yin J. 2012. Bulk universality for generalized Wigner matrices. Probability Theory and Related Fields. 154(1–2), 341–407.","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."},"extern":1,"type":"journal_article","status":"public","_id":"2767","page":"341 - 407","date_created":"2018-12-11T11:59:29Z","volume":154,"date_published":"2012-10-01T00:00:00Z","doi":"10.1007/s00440-011-0390-3","issue":"1-2","year":"2012","publication_status":"published","publication":"Probability Theory and Related Fields","day":"01","quality_controlled":0,"publisher":"Springer","intvolume":" 154","month":"10","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. "}]},{"date_created":"2018-12-11T11:59:30Z","issue":"2","date_published":"2012-01-01T00:00:00Z","doi":"10.1007/s00220-011-1373-z","volume":309,"page":"507 - 542","publication":"Communications in Mathematical Physics","day":"01","year":"2012","publication_status":"published","intvolume":" 309","month":"01","quality_controlled":0,"publisher":"Springer","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."}],"title":"Wegner estimate and Anderson localization for random magnetic fields","publist_id":"4122","author":[{"last_name":"Erdös","full_name":"László Erdös","orcid":"0000-0001-5366-9603","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Hasler","full_name":"Hasler, David G","first_name":"David"}],"extern":1,"date_updated":"2021-01-12T06:59:34Z","citation":{"ista":"Erdös L, Hasler D. 2012. Wegner estimate and Anderson localization for random magnetic fields. Communications in Mathematical Physics. 309(2), 507–542.","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.","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","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","short":"L. Erdös, D. Hasler, Communications in Mathematical Physics 309 (2012) 507–542.","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.","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."},"status":"public","type":"journal_article","_id":"2768"},{"abstract":[{"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.","lang":"eng"}],"intvolume":" 49","month":"01","quality_controlled":0,"publisher":"American Mathematical Society","publication":"Bulletin of the American Mathematical Society","day":"30","year":"2012","publication_status":"published","date_created":"2018-12-11T11:59:32Z","issue":"3","doi":"10.1090/S0273-0979-2012-01372-1","date_published":"2012-01-30T00:00:00Z","volume":49,"page":"377 - 414","_id":"2775","status":"public","type":"journal_article","extern":1,"citation":{"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.","short":"L. Erdös, H. Yau, Bulletin of the American Mathematical Society 49 (2012) 377–414.","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","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","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.","ista":"Erdös L, Yau H. 2012. Universality of local spectral statistics of random matrices. Bulletin of the American Mathematical Society. 49(3), 377–414.","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."},"date_updated":"2021-01-12T06:59:36Z","title":"Universality of local spectral statistics of random matrices","publist_id":"4115","author":[{"first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","last_name":"Erdös","full_name":"László Erdös","orcid":"0000-0001-5366-9603"},{"first_name":"Horng","full_name":"Yau, Horng-Tzer","last_name":"Yau"}]},{"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."}],"quality_controlled":0,"publisher":"American Institute of Physics","intvolume":" 53","month":"09","year":"2012","publication_status":"published","publication":"Journal of Mathematical Physics","day":"28","date_created":"2018-12-11T11:59:32Z","volume":53,"issue":"9","date_published":"2012-09-28T00:00:00Z","doi":"10.1063/1.3697417","_id":"2777","type":"journal_article","status":"public","citation":{"ista":"Erdös L, Fournais S, Solovej J. 2012. Relativistic Scott correction in self-generated magnetic fields. Journal of Mathematical Physics. 53(9).","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.","short":"L. Erdös, S. Fournais, J. Solovej, Journal of Mathematical Physics 53 (2012).","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","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."},"date_updated":"2021-01-12T06:59:37Z","extern":1,"author":[{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","last_name":"Erdös","orcid":"0000-0001-5366-9603","full_name":"László Erdös"},{"last_name":"Fournais","full_name":"Fournais, Søren","first_name":"Søren"},{"last_name":"Solovej","full_name":"Solovej, Jan P","first_name":"Jan"}],"publist_id":"4113","title":"Relativistic Scott correction in self-generated magnetic fields"},{"abstract":[{"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.","lang":"eng"}],"publisher":"Birkhäuser","quality_controlled":0,"month":"05","intvolume":" 13","year":"2012","publication_status":"published","day":"01","publication":"Annales Henri Poincare","page":"671 - 730","issue":"4","volume":13,"doi":"10.1007/s00023-011-0150-z","date_published":"2012-05-01T00:00:00Z","date_created":"2018-12-11T11:59:31Z","_id":"2772","type":"journal_article","status":"public","citation":{"ista":"Erdös L, Fournais S, Solovej J. 2012. Second order semiclassics with self generated magnetic fields. Annales Henri Poincare. 13(4), 671–730.","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.","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","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","short":"L. Erdös, S. Fournais, J. Solovej, Annales Henri Poincare 13 (2012) 671–730.","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.","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."},"date_updated":"2021-01-12T06:59:36Z","extern":1,"publist_id":"4118","author":[{"orcid":"0000-0001-5366-9603","full_name":"László Erdös","last_name":"Erdös","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Søren","full_name":"Fournais, Søren","last_name":"Fournais"},{"first_name":"Jan","last_name":"Solovej","full_name":"Solovej, Jan P"}],"title":"Second order semiclassics with self generated magnetic fields"},{"status":"public","type":"journal_article","_id":"2776","title":"Spectral statistics of Erdős-Rényi graphs II: Eigenvalue spacing and the extreme eigenvalues","author":[{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","last_name":"Erdös","full_name":"László Erdös","orcid":"0000-0001-5366-9603"},{"last_name":"Knowles","full_name":"Knowles, Antti","first_name":"Antti"},{"first_name":"Horng","last_name":"Yau","full_name":"Yau, Horng-Tzer"},{"first_name":"Jun","full_name":"Yin, Jun","last_name":"Yin"}],"publist_id":"4114","extern":1,"citation":{"ista":"Erdös L, Knowles A, Yau H, Yin J. 2012. Spectral statistics of Erdős-Rényi graphs II: Eigenvalue spacing and the extreme eigenvalues. Communications in Mathematical Physics. 314(3), 587–640.","chicago":"Erdös, László, Antti Knowles, Horng Yau, and Jun Yin. “Spectral Statistics of Erdős-Rényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues.” Communications in Mathematical Physics. Springer, 2012. https://doi.org/10.1007/s00220-012-1527-7.","ieee":"L. Erdös, A. Knowles, H. Yau, and J. Yin, “Spectral statistics of Erdős-Rényi graphs II: Eigenvalue spacing and the extreme eigenvalues,” Communications in Mathematical Physics, vol. 314, no. 3. Springer, pp. 587–640, 2012.","short":"L. Erdös, A. Knowles, H. Yau, J. Yin, Communications in Mathematical Physics 314 (2012) 587–640.","apa":"Erdös, L., Knowles, A., Yau, H., & Yin, J. (2012). Spectral statistics of Erdős-Rényi graphs II: Eigenvalue spacing and the extreme eigenvalues. Communications in Mathematical Physics. Springer. https://doi.org/10.1007/s00220-012-1527-7","ama":"Erdös L, Knowles A, Yau H, Yin J. Spectral statistics of Erdős-Rényi graphs II: Eigenvalue spacing and the extreme eigenvalues. Communications in Mathematical Physics. 2012;314(3):587-640. doi:10.1007/s00220-012-1527-7","mla":"Erdös, László, et al. “Spectral Statistics of Erdős-Rényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues.” Communications in Mathematical Physics, vol. 314, no. 3, Springer, 2012, pp. 587–640, doi:10.1007/s00220-012-1527-7."},"date_updated":"2021-01-12T06:59:37Z","intvolume":" 314","month":"09","publisher":"Springer","quality_controlled":0,"abstract":[{"lang":"eng","text":"We consider the ensemble of adjacency matrices of Erdős-Rényi random graphs, i.e. graphs on N vertices where every edge is chosen independently and with probability p ≡ p(N). We rescale the matrix so that its bulk eigenvalues are of order one. Under the assumption pN≫N2/3 , we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum. More precisely, we prove (1) that the eigenvalue spacing of the Erdős-Rényi graph in the bulk of the spectrum has the same distribution as that of the Gaussian orthogonal ensemble; and (2) that the second largest eigenvalue of the Erdős-Rényi graph has the same distribution as the largest eigenvalue of the Gaussian orthogonal ensemble. As an application of our method, we prove the bulk universality of generalized Wigner matrices under the assumption that the matrix entries have at least 4 + ε moments."}],"date_created":"2018-12-11T11:59:32Z","issue":"3","volume":314,"date_published":"2012-09-01T00:00:00Z","doi":"10.1007/s00220-012-1527-7","page":"587 - 640","publication":"Communications in Mathematical Physics","day":"01","publication_status":"published","year":"2012"},{"abstract":[{"text":"We consider a large neutral molecule with total nuclear charge Z in non-relativistic quantum mechanics with a self-generated classical electromagnetic field. To ensure stability, we assume that Zα 2 ≤ κ 0 for a sufficiently small κ 0, where α denotes the fine structure constant. We show that, in the simultaneous limit Z → ∞, α → 0 such that κ = Zα 2 is fixed, the ground state energy of the system is given by a two term expansion c 1Z 7/3 + c 2(κ) Z 2 + o(Z 2). The leading term is given by the non-magnetic Thomas-Fermi theory. Our result shows that the magnetic field affects only the second (so-called Scott) term in the expansion.","lang":"eng"}],"quality_controlled":0,"publisher":"Springer","month":"06","intvolume":" 312","year":"2012","publication_status":"published","day":"01","publication":"Communications in Mathematical Physics","page":"847 - 882","date_published":"2012-06-01T00:00:00Z","issue":"3","volume":312,"doi":"10.1007/s00220-012-1468-1","date_created":"2018-12-11T11:59:31Z","_id":"2774","type":"journal_article","status":"public","date_updated":"2021-01-12T06:59:36Z","citation":{"ista":"Erdös L, Fournais S, Solovej J. 2012. Scott correction for large atoms and molecules in a self-generated magnetic field. Communications in Mathematical Physics. 312(3), 847–882.","chicago":"Erdös, László, Søren Fournais, and Jan Solovej. “Scott Correction for Large Atoms and Molecules in a Self-Generated Magnetic Field.” Communications in Mathematical Physics. Springer, 2012. https://doi.org/10.1007/s00220-012-1468-1.","short":"L. Erdös, S. Fournais, J. Solovej, Communications in Mathematical Physics 312 (2012) 847–882.","ieee":"L. Erdös, S. Fournais, and J. Solovej, “Scott correction for large atoms and molecules in a self-generated magnetic field,” Communications in Mathematical Physics, vol. 312, no. 3. Springer, pp. 847–882, 2012.","ama":"Erdös L, Fournais S, Solovej J. Scott correction for large atoms and molecules in a self-generated magnetic field. Communications in Mathematical Physics. 2012;312(3):847-882. doi:10.1007/s00220-012-1468-1","apa":"Erdös, L., Fournais, S., & Solovej, J. (2012). Scott correction for large atoms and molecules in a self-generated magnetic field. Communications in Mathematical Physics. Springer. https://doi.org/10.1007/s00220-012-1468-1","mla":"Erdös, László, et al. “Scott Correction for Large Atoms and Molecules in a Self-Generated Magnetic Field.” Communications in Mathematical Physics, vol. 312, no. 3, Springer, 2012, pp. 847–82, doi:10.1007/s00220-012-1468-1."},"extern":1,"author":[{"first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","last_name":"Erdös","orcid":"0000-0001-5366-9603","full_name":"László Erdös"},{"first_name":"Søren","last_name":"Fournais","full_name":"Fournais, Søren"},{"first_name":"Jan","last_name":"Solovej","full_name":"Solovej, Jan P"}],"publist_id":"4116","title":"Scott correction for large atoms and molecules in a self-generated magnetic field"}]