[{"external_id":{"isi":["000356993100012"],"arxiv":["1208.5823"]},"title":"The logarithmic law of random determinant","publication_status":"published","publication":"Bernoulli","arxiv":1,"department":[{"_id":"LaEr"}],"doi":"10.3150/14-BEJ615","volume":21,"main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1208.5823"}],"language":[{"iso":"eng"}],"oa_version":"Preprint","citation":{"short":"Z. Bao, G. Pan, W. Zhou, Bernoulli 21 (2015) 1600–1628.","mla":"Bao, Zhigang, et al. “The Logarithmic Law of Random Determinant.” <i>Bernoulli</i>, vol. 21, no. 3, Bernoulli Society for Mathematical Statistics and Probability, 2015, pp. 1600–28, doi:<a href=\"https://doi.org/10.3150/14-BEJ615\">10.3150/14-BEJ615</a>.","apa":"Bao, Z., Pan, G., &#38; Zhou, W. (2015). The logarithmic law of random determinant. <i>Bernoulli</i>. Bernoulli Society for Mathematical Statistics and Probability. <a href=\"https://doi.org/10.3150/14-BEJ615\">https://doi.org/10.3150/14-BEJ615</a>","chicago":"Bao, Zhigang, Guangming Pan, and Wang Zhou. “The Logarithmic Law of Random Determinant.” <i>Bernoulli</i>. Bernoulli Society for Mathematical Statistics and Probability, 2015. <a href=\"https://doi.org/10.3150/14-BEJ615\">https://doi.org/10.3150/14-BEJ615</a>.","ama":"Bao Z, Pan G, Zhou W. The logarithmic law of random determinant. <i>Bernoulli</i>. 2015;21(3):1600-1628. doi:<a href=\"https://doi.org/10.3150/14-BEJ615\">10.3150/14-BEJ615</a>","ieee":"Z. Bao, G. Pan, and W. Zhou, “The logarithmic law of random determinant,” <i>Bernoulli</i>, vol. 21, no. 3. Bernoulli Society for Mathematical Statistics and Probability, pp. 1600–1628, 2015.","ista":"Bao Z, Pan G, Zhou W. 2015. The logarithmic law of random determinant. Bernoulli. 21(3), 1600–1628."},"fulldoi":"https://doi.org/10.3150/14-BEJ615","publisher":"Bernoulli Society for Mathematical Statistics and Probability","year":"2015","publist_id":"5671","abstract":[{"text":"Consider the square random matrix An = (aij)n,n, where {aij:= a(n)ij , i, j = 1, . . . , n} is a collection of independent real random variables with means zero and variances one. Under the additional moment condition supn max1≤i,j ≤n Ea4ij &lt;∞, we prove Girko's logarithmic law of det An in the sense that as n→∞ log | detAn| ? (1/2) log(n-1)! d/→√(1/2) log n N(0, 1).","lang":"eng"}],"intvolume":"        21","oa":1,"quality_controlled":"1","day":"01","date_published":"2015-08-01T00:00:00Z","date_updated":"2025-09-23T13:59:56Z","month":"08","_id":"1506","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_created":"2018-12-11T11:52:25Z","article_processing_charge":"No","status":"public","type":"journal_article","isi":1,"issue":"3","author":[{"id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","first_name":"Zhigang","orcid":"0000-0003-3036-1475","full_name":"Bao, Zhigang","last_name":"Bao"},{"full_name":"Pan, Guangming","last_name":"Pan","first_name":"Guangming"},{"full_name":"Zhou, Wang","last_name":"Zhou","first_name":"Wang"}],"page":"1600 - 1628"},{"department":[{"_id":"LaEr"}],"doi":"10.1109/TIT.2015.2421894","publication_status":"published","title":"Asymptotic mutual information statistics of MIMO channels and CLT of sample covariance matrices","external_id":{"isi":["000354943600029"]},"publication":"IEEE Transactions on Information Theory","year":"2015","publisher":"IEEE","language":[{"iso":"eng"}],"volume":61,"fulldoi":"https://doi.org/10.1109/TIT.2015.2421894","scopus_import":"1","citation":{"chicago":"Bao, Zhigang, Guangming Pan, and Wang Zhou. “Asymptotic Mutual Information Statistics of MIMO Channels and CLT of Sample Covariance Matrices.” <i>IEEE Transactions on Information Theory</i>. IEEE, 2015. <a href=\"https://doi.org/10.1109/TIT.2015.2421894\">https://doi.org/10.1109/TIT.2015.2421894</a>.","ama":"Bao Z, Pan G, Zhou W. Asymptotic mutual information statistics of MIMO channels and CLT of sample covariance matrices. <i>IEEE Transactions on Information Theory</i>. 2015;61(6):3413-3426. doi:<a href=\"https://doi.org/10.1109/TIT.2015.2421894\">10.1109/TIT.2015.2421894</a>","ieee":"Z. Bao, G. Pan, and W. Zhou, “Asymptotic mutual information statistics of MIMO channels and CLT of sample covariance matrices,” <i>IEEE Transactions on Information Theory</i>, vol. 61, no. 6. IEEE, pp. 3413–3426, 2015.","ista":"Bao Z, Pan G, Zhou W. 2015. Asymptotic mutual information statistics of MIMO channels and CLT of sample covariance matrices. IEEE Transactions on Information Theory. 61(6), 3413–3426.","short":"Z. Bao, G. Pan, W. Zhou, IEEE Transactions on Information Theory 61 (2015) 3413–3426.","mla":"Bao, Zhigang, et al. “Asymptotic Mutual Information Statistics of MIMO Channels and CLT of Sample Covariance Matrices.” <i>IEEE Transactions on Information Theory</i>, vol. 61, no. 6, IEEE, 2015, pp. 3413–26, doi:<a href=\"https://doi.org/10.1109/TIT.2015.2421894\">10.1109/TIT.2015.2421894</a>.","apa":"Bao, Z., Pan, G., &#38; Zhou, W. (2015). Asymptotic mutual information statistics of MIMO channels and CLT of sample covariance matrices. <i>IEEE Transactions on Information Theory</i>. IEEE. <a href=\"https://doi.org/10.1109/TIT.2015.2421894\">https://doi.org/10.1109/TIT.2015.2421894</a>"},"oa_version":"None","date_published":"2015-06-01T00:00:00Z","quality_controlled":"1","day":"01","month":"06","date_updated":"2025-09-23T07:57:31Z","publist_id":"5586","acknowledgement":"G. Pan was supported by MOE Tier 2 under Grant 2014-T2-2-060 and in part by Tier 1 under Grant RG25/14 through the Nanyang Technological University, Singapore. W. Zhou was supported by the National University of Singapore, Singapore, under Grant R-155-000-131-112.\r\n","intvolume":"        61","abstract":[{"lang":"eng","text":"In this paper, we consider the fluctuation of mutual information statistics of a multiple input multiple output channel communication systems without assuming that the entries of the channel matrix have zero pseudovariance. To this end, we also establish a central limit theorem of the linear spectral statistics for sample covariance matrices under general moment conditions by removing the restrictions imposed on the second moment and fourth moment on the matrix entries in Bai and Silverstein (2004)."}],"isi":1,"issue":"6","type":"journal_article","page":"3413 - 3426","author":[{"last_name":"Bao","full_name":"Bao, Zhigang","orcid":"0000-0003-3036-1475","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","first_name":"Zhigang"},{"last_name":"Pan","full_name":"Pan, Guangming","first_name":"Guangming"},{"first_name":"Wang","full_name":"Zhou, Wang","last_name":"Zhou"}],"date_created":"2018-12-11T11:52:52Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","_id":"1585","status":"public","article_processing_charge":"No"},{"article_number":"1550018","publication":"Reviews in Mathematical Physics","title":"Edge universality for deformed Wigner matrices","publication_status":"published","external_id":{"arxiv":["1407.8015"],"isi":["000362566600001"]},"doi":"10.1142/S0129055X1550018X","department":[{"_id":"LaEr"}],"arxiv":1,"fulldoi":"https://doi.org/10.1142/S0129055X1550018X","scopus_import":"1","citation":{"apa":"Lee, J., &#38; Schnelli, K. (2015). Edge universality for deformed Wigner matrices. <i>Reviews in Mathematical Physics</i>. World Scientific Publishing. <a href=\"https://doi.org/10.1142/S0129055X1550018X\">https://doi.org/10.1142/S0129055X1550018X</a>","mla":"Lee, Jioon, and Kevin Schnelli. “Edge Universality for Deformed Wigner Matrices.” <i>Reviews in Mathematical Physics</i>, vol. 27, no. 8, 1550018, World Scientific Publishing, 2015, doi:<a href=\"https://doi.org/10.1142/S0129055X1550018X\">10.1142/S0129055X1550018X</a>.","short":"J. Lee, K. Schnelli, Reviews in Mathematical Physics 27 (2015).","ieee":"J. Lee and K. Schnelli, “Edge universality for deformed Wigner matrices,” <i>Reviews in Mathematical Physics</i>, vol. 27, no. 8. World Scientific Publishing, 2015.","ista":"Lee J, Schnelli K. 2015. Edge universality for deformed Wigner matrices. Reviews in Mathematical Physics. 27(8), 1550018.","ama":"Lee J, Schnelli K. Edge universality for deformed Wigner matrices. <i>Reviews in Mathematical Physics</i>. 2015;27(8). doi:<a href=\"https://doi.org/10.1142/S0129055X1550018X\">10.1142/S0129055X1550018X</a>","chicago":"Lee, Jioon, and Kevin Schnelli. “Edge Universality for Deformed Wigner Matrices.” <i>Reviews in Mathematical Physics</i>. World Scientific Publishing, 2015. <a href=\"https://doi.org/10.1142/S0129055X1550018X\">https://doi.org/10.1142/S0129055X1550018X</a>."},"oa_version":"Preprint","language":[{"iso":"eng"}],"volume":27,"main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1407.8015"}],"publisher":"World Scientific Publishing","year":"2015","abstract":[{"lang":"eng","text":"We consider N × N random matrices of the form H = W + V where W is a real symmetric Wigner matrix and V a random or deterministic, real, diagonal matrix whose entries are independent of W. We assume subexponential decay for the matrix entries of W and we choose V so that the eigenvalues of W and V are typically of the same order. For a large class of diagonal matrices V, we show that the rescaled distribution of the extremal eigenvalues is given by the Tracy-Widom distribution F1 in the limit of large N. Our proofs also apply to the complex Hermitian setting, i.e. when W is a complex Hermitian Wigner matrix."}],"intvolume":"        27","publist_id":"5475","month":"09","date_updated":"2025-09-23T08:32:19Z","date_published":"2015-09-01T00:00:00Z","oa":1,"day":"01","quality_controlled":"1","status":"public","article_processing_charge":"No","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_created":"2018-12-11T11:53:24Z","_id":"1674","author":[{"first_name":"Jioon","full_name":"Lee, Jioon","last_name":"Lee"},{"id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","first_name":"Kevin","orcid":"0000-0003-0954-3231","full_name":"Schnelli, Kevin","last_name":"Schnelli"}],"issue":"8","isi":1,"type":"journal_article"},{"_id":"1864","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2018-12-11T11:54:26Z","article_processing_charge":"No","status":"public","type":"journal_article","issue":"3","isi":1,"author":[{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","full_name":"Erdös, László","orcid":"0000-0001-5366-9603","last_name":"Erdös"},{"first_name":"Antti","full_name":"Knowles, Antti","last_name":"Knowles"}],"page":"709 - 799","publist_id":"5233","abstract":[{"lang":"eng","text":"The Altshuler–Shklovskii formulas (Altshuler and Shklovskii, BZh Eksp Teor Fiz 91:200, 1986) predict, for any disordered quantum system in the diffusive regime, a universal power law behaviour for the correlation functions of the mesoscopic eigenvalue density. In this paper and its companion (Erdős and Knowles, The Altshuler–Shklovskii formulas for random band matrices I: the unimodular case, 2013), we prove these formulas for random band matrices. In (Erdős and Knowles, The Altshuler–Shklovskii formulas for random band matrices I: the unimodular case, 2013) we introduced a diagrammatic approach and presented robust estimates on general diagrams under certain simplifying assumptions. In this paper, we remove these assumptions by giving a general estimate of the subleading diagrams. We also give a precise analysis of the leading diagrams which give rise to the Altschuler–Shklovskii power laws. Moreover, we introduce a family of general random band matrices which interpolates between real symmetric (β = 1) and complex Hermitian (β = 2) models, and track the transition for the mesoscopic density–density correlation. Finally, we address the higher-order correlation functions by proving that they behave asymptotically according to a Gaussian process whose covariance is given by the Altshuler–Shklovskii formulas.\r\n"}],"intvolume":"        16","oa":1,"day":"01","date_published":"2015-03-01T00:00:00Z","date_updated":"2025-10-22T10:23:38Z","month":"03","volume":16,"main_file_link":[{"url":"http://arxiv.org/abs/1309.5107","open_access":"1"}],"language":[{"iso":"eng"}],"citation":{"short":"L. Erdös, A. Knowles, Annales Henri Poincare 16 (2015) 709–799.","apa":"Erdös, L., &#38; Knowles, A. (2015). The Altshuler-Shklovskii formulas for random band matrices II: The general case. <i>Annales Henri Poincare</i>. Springer. <a href=\"https://doi.org/10.1007/s00023-014-0333-5\">https://doi.org/10.1007/s00023-014-0333-5</a>","mla":"Erdös, László, and Antti Knowles. “The Altshuler-Shklovskii Formulas for Random Band Matrices II: The General Case.” <i>Annales Henri Poincare</i>, vol. 16, no. 3, Springer, 2015, pp. 709–99, doi:<a href=\"https://doi.org/10.1007/s00023-014-0333-5\">10.1007/s00023-014-0333-5</a>.","ista":"Erdös L, Knowles A. 2015. The Altshuler-Shklovskii formulas for random band matrices II: The general case. Annales Henri Poincare. 16(3), 709–799.","ieee":"L. Erdös and A. Knowles, “The Altshuler-Shklovskii formulas for random band matrices II: The general case,” <i>Annales Henri Poincare</i>, vol. 16, no. 3. Springer, pp. 709–799, 2015.","ama":"Erdös L, Knowles A. The Altshuler-Shklovskii formulas for random band matrices II: The general case. <i>Annales Henri Poincare</i>. 2015;16(3):709-799. doi:<a href=\"https://doi.org/10.1007/s00023-014-0333-5\">10.1007/s00023-014-0333-5</a>","chicago":"Erdös, László, and Antti Knowles. “The Altshuler-Shklovskii Formulas for Random Band Matrices II: The General Case.” <i>Annales Henri Poincare</i>. Springer, 2015. <a href=\"https://doi.org/10.1007/s00023-014-0333-5\">https://doi.org/10.1007/s00023-014-0333-5</a>."},"scopus_import":"1","oa_version":"Preprint","fulldoi":"https://doi.org/10.1007/s00023-014-0333-5","project":[{"grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems"}],"publisher":"Springer","year":"2015","external_id":{"arxiv":["1309.5107"],"isi":["000349364100002"]},"title":"The Altshuler-Shklovskii formulas for random band matrices II: The general case","publication_status":"published","publication":"Annales Henri Poincare","arxiv":1,"ec_funded":1,"department":[{"_id":"LaEr"}],"doi":"10.1007/s00023-014-0333-5"},{"publist_id":"5282","intvolume":"         6","abstract":[{"text":"Condensation phenomena arise through a collective behaviour of particles. They are observed in both classical and quantum systems, ranging from the formation of traffic jams in mass transport models to the macroscopic occupation of the energetic ground state in ultra-cold bosonic gases (Bose-Einstein condensation). Recently, it has been shown that a driven and dissipative system of bosons may form multiple condensates. Which states become the condensates has, however, remained elusive thus far. The dynamics of this condensation are described by coupled birth-death processes, which also occur in evolutionary game theory. Here we apply concepts from evolutionary game theory to explain the formation of multiple condensates in such driven-dissipative bosonic systems. We show that the vanishing of relative entropy production determines their selection. The condensation proceeds exponentially fast, but the system never comes to rest. Instead, the occupation numbers of condensates may oscillate, as we demonstrate for a rock-paper-scissors game of condensates.","lang":"eng"}],"file_date_updated":"2020-07-14T12:45:17Z","day":"24","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"quality_controlled":"1","oa":1,"date_published":"2015-04-24T00:00:00Z","date_updated":"2025-09-23T08:44:01Z","month":"04","_id":"1824","pubrep_id":"451","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_created":"2018-12-11T11:54:13Z","article_processing_charge":"No","status":"public","isi":1,"type":"journal_article","author":[{"last_name":"Knebel","full_name":"Knebel, Johannes","first_name":"Johannes"},{"last_name":"Weber","full_name":"Weber, Markus","first_name":"Markus"},{"first_name":"Torben H","id":"3020C786-F248-11E8-B48F-1D18A9856A87","last_name":"Krüger","full_name":"Krüger, Torben H","orcid":"0000-0002-4821-3297"},{"first_name":"Erwin","last_name":"Frey","full_name":"Frey, Erwin"}],"external_id":{"isi":["000353705000003"]},"publication_status":"published","title":"Evolutionary games of condensates in coupled birth-death processes","publication":"Nature Communications","article_number":"6977","department":[{"_id":"LaEr"}],"has_accepted_license":"1","doi":"10.1038/ncomms7977","ddc":["530"],"volume":6,"language":[{"iso":"eng"}],"scopus_import":"1","citation":{"short":"J. Knebel, M. Weber, T.H. Krüger, E. Frey, Nature Communications 6 (2015).","apa":"Knebel, J., Weber, M., Krüger, T. H., &#38; Frey, E. (2015). Evolutionary games of condensates in coupled birth-death processes. <i>Nature Communications</i>. Nature Publishing Group. <a href=\"https://doi.org/10.1038/ncomms7977\">https://doi.org/10.1038/ncomms7977</a>","mla":"Knebel, Johannes, et al. “Evolutionary Games of Condensates in Coupled Birth-Death Processes.” <i>Nature Communications</i>, vol. 6, 6977, Nature Publishing Group, 2015, doi:<a href=\"https://doi.org/10.1038/ncomms7977\">10.1038/ncomms7977</a>.","ista":"Knebel J, Weber M, Krüger TH, Frey E. 2015. Evolutionary games of condensates in coupled birth-death processes. Nature Communications. 6, 6977.","ieee":"J. Knebel, M. Weber, T. H. Krüger, and E. Frey, “Evolutionary games of condensates in coupled birth-death processes,” <i>Nature Communications</i>, vol. 6. Nature Publishing Group, 2015.","ama":"Knebel J, Weber M, Krüger TH, Frey E. Evolutionary games of condensates in coupled birth-death processes. <i>Nature Communications</i>. 2015;6. doi:<a href=\"https://doi.org/10.1038/ncomms7977\">10.1038/ncomms7977</a>","chicago":"Knebel, Johannes, Markus Weber, Torben H Krüger, and Erwin Frey. “Evolutionary Games of Condensates in Coupled Birth-Death Processes.” <i>Nature Communications</i>. Nature Publishing Group, 2015. <a href=\"https://doi.org/10.1038/ncomms7977\">https://doi.org/10.1038/ncomms7977</a>."},"oa_version":"Published Version","fulldoi":"https://doi.org/10.1038/ncomms7977","file":[{"file_id":"5245","content_type":"application/pdf","creator":"system","relation":"main_file","checksum":"c4cffb5c8b245e658a34eac71a03e7cc","file_size":1151501,"file_name":"IST-2016-451-v1+1_ncomms7977.pdf","date_updated":"2020-07-14T12:45:17Z","date_created":"2018-12-12T10:16:54Z","access_level":"open_access"}],"year":"2015","publisher":"Nature Publishing Group"},{"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2018-12-11T11:52:26Z","_id":"1508","status":"public","article_processing_charge":"No","isi":1,"type":"journal_article","issue":"8","page":"1927 - 2036","author":[{"first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","last_name":"Erdös","full_name":"Erdös, László","orcid":"0000-0001-5366-9603"},{"first_name":"Horng","last_name":"Yau","full_name":"Yau, Horng"}],"publist_id":"5669","abstract":[{"text":"We consider generalized Wigner ensembles and general β-ensembles with analytic potentials for any β ≥ 1. The recent universality results in particular assert that the local averages of consecutive eigenvalue gaps in the bulk of the spectrum are universal in the sense that they coincide with those of the corresponding Gaussian β-ensembles. In this article, we show that local averaging is not necessary for this result, i.e. we prove that the single gap distributions in the bulk are universal. In fact, with an additional step, our result can be extended to any C4(ℝ) potential.","lang":"eng"}],"intvolume":"        17","date_published":"2015-08-01T00:00:00Z","oa":1,"day":"01","quality_controlled":"1","month":"08","date_updated":"2026-07-06T11:55:30Z","language":[{"iso":"eng"}],"volume":17,"main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1211.3786"}],"fulldoi":"https://doi.org/10.4171/JEMS/548","citation":{"short":"L. Erdös, H. Yau, Journal of the European Mathematical Society 17 (2015) 1927–2036.","apa":"Erdös, L., &#38; Yau, H. (2015). Gap universality of generalized Wigner and β ensembles. <i>Journal of the European Mathematical Society</i>. EMS Press. <a href=\"https://doi.org/10.4171/JEMS/548\">https://doi.org/10.4171/JEMS/548</a>","mla":"Erdös, László, and Horng Yau. “Gap Universality of Generalized Wigner and β Ensembles.” <i>Journal of the European Mathematical Society</i>, vol. 17, no. 8, EMS Press, 2015, pp. 1927–2036, doi:<a href=\"https://doi.org/10.4171/JEMS/548\">10.4171/JEMS/548</a>.","ieee":"L. Erdös and H. Yau, “Gap universality of generalized Wigner and β ensembles,” <i>Journal of the European Mathematical Society</i>, vol. 17, no. 8. EMS Press, pp. 1927–2036, 2015.","ista":"Erdös L, Yau H. 2015. Gap universality of generalized Wigner and β ensembles. Journal of the European Mathematical Society. 17(8), 1927–2036.","ama":"Erdös L, Yau H. Gap universality of generalized Wigner and β ensembles. <i>Journal of the European Mathematical Society</i>. 2015;17(8):1927-2036. doi:<a href=\"https://doi.org/10.4171/JEMS/548\">10.4171/JEMS/548</a>","chicago":"Erdös, László, and Horng Yau. “Gap Universality of Generalized Wigner and β Ensembles.” <i>Journal of the European Mathematical Society</i>. EMS Press, 2015. <a href=\"https://doi.org/10.4171/JEMS/548\">https://doi.org/10.4171/JEMS/548</a>."},"scopus_import":"1","oa_version":"Preprint","publisher":"EMS Press","year":"2015","title":"Gap universality of generalized Wigner and β ensembles","das_tickbox":"1","publication_status":"published","external_id":{"isi":["000360822900003"],"arxiv":["1211.3786"]},"publication":"Journal of the European Mathematical Society","department":[{"_id":"LaEr"}],"arxiv":1,"doi":"10.4171/JEMS/548"},{"day":"09","quality_controlled":"1","oa":1,"date_published":"2015-10-09T00:00:00Z","date_updated":"2026-08-12T13:58:57Z","month":"10","publist_id":"5472","intvolume":"        56","abstract":[{"text":"We consider real symmetric and complex Hermitian random matrices with the additional symmetry hxy = hN-y,N-x. The matrix elements are independent (up to the fourfold symmetry) and not necessarily identically distributed. This ensemble naturally arises as the Fourier transform of a Gaussian orthogonal ensemble. Italso occurs as the flip matrix model - an approximation of the two-dimensional Anderson model at small disorder. We show that the density of states converges to the Wigner semicircle law despite the new symmetry type. We also prove the local version of the semicircle law on the optimal scale.","lang":"eng"}],"isi":1,"issue":"10","type":"journal_article","author":[{"full_name":"Alt, Johannes","last_name":"Alt","id":"36D3D8B6-F248-11E8-B48F-1D18A9856A87","first_name":"Johannes"}],"corr_author":"1","_id":"1677","related_material":{"record":[{"status":"public","id":"149","relation":"dissertation_contains"}]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2018-12-11T11:53:25Z","article_processing_charge":"No","status":"public","ec_funded":1,"arxiv":1,"department":[{"_id":"LaEr"}],"doi":"10.1063/1.4932606","external_id":{"isi":["000364237000026"],"arxiv":["1506.04683"]},"publication_status":"published","title":"The local semicircle law for random matrices with a fourfold symmetry","publication":"Journal of Mathematical Physics","article_number":"103301","year":"2015","publisher":"AIP Publishing","main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1506.04683"}],"volume":56,"language":[{"iso":"eng"}],"citation":{"mla":"Alt, Johannes. “The Local Semicircle Law for Random Matrices with a Fourfold Symmetry.” <i>Journal of Mathematical Physics</i>, vol. 56, no. 10, 103301, AIP Publishing, 2015, doi:<a href=\"https://doi.org/10.1063/1.4932606\">10.1063/1.4932606</a>.","apa":"Alt, J. (2015). The local semicircle law for random matrices with a fourfold symmetry. <i>Journal of Mathematical Physics</i>. AIP Publishing. <a href=\"https://doi.org/10.1063/1.4932606\">https://doi.org/10.1063/1.4932606</a>","short":"J. Alt, Journal of Mathematical Physics 56 (2015).","ama":"Alt J. The local semicircle law for random matrices with a fourfold symmetry. <i>Journal of Mathematical Physics</i>. 2015;56(10). doi:<a href=\"https://doi.org/10.1063/1.4932606\">10.1063/1.4932606</a>","chicago":"Alt, Johannes. “The Local Semicircle Law for Random Matrices with a Fourfold Symmetry.” <i>Journal of Mathematical Physics</i>. AIP Publishing, 2015. <a href=\"https://doi.org/10.1063/1.4932606\">https://doi.org/10.1063/1.4932606</a>.","ieee":"J. Alt, “The local semicircle law for random matrices with a fourfold symmetry,” <i>Journal of Mathematical Physics</i>, vol. 56, no. 10. AIP Publishing, 2015.","ista":"Alt J. 2015. The local semicircle law for random matrices with a fourfold symmetry. Journal of Mathematical Physics. 56(10), 103301."},"scopus_import":"1","oa_version":"Preprint","fulldoi":"https://doi.org/10.1063/1.4932606","project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804","name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7"}]},{"publist_id":"4739","file_date_updated":"2020-07-14T12:45:34Z","abstract":[{"lang":"eng","text":"We consider sample covariance matrices of the form X∗X, where X is an M×N matrix with independent random entries.  We prove the isotropic local Marchenko-Pastur law, i.e. we prove that the resolvent (X∗X−z)−1 converges to a multiple of the identity in the sense of quadratic forms. More precisely, we establish sharp high-probability bounds on the quantity ⟨v,(X∗X−z)−1w⟩−⟨v,w⟩m(z), where m is the Stieltjes transform of the Marchenko-Pastur law and v,w∈CN. We require the logarithms of the dimensions M and N to be comparable. Our result holds down to scales Iz≥N−1+ε and throughout the entire spectrum away from 0. We also prove analogous results for generalized Wigner matrices.\r\n"}],"intvolume":"        19","date_published":"2014-03-15T00:00:00Z","oa":1,"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"day":"15","quality_controlled":"1","month":"03","date_updated":"2025-09-29T11:26:42Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_created":"2018-12-11T11:56:25Z","pubrep_id":"427","_id":"2225","status":"public","article_processing_charge":"No","isi":1,"type":"journal_article","author":[{"full_name":"Bloemendal, Alex","last_name":"Bloemendal","first_name":"Alex"},{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","orcid":"0000-0001-5366-9603","full_name":"Erdös, László","last_name":"Erdös"},{"first_name":"Antti","full_name":"Knowles, Antti","last_name":"Knowles"},{"first_name":"Horng","last_name":"Yau","full_name":"Yau, Horng"},{"first_name":"Jun","last_name":"Yin","full_name":"Yin, Jun"}],"title":"Isotropic local laws for sample covariance and generalized Wigner matrices","publication_status":"published","external_id":{"isi":["000340998500001"]},"article_number":"33","publication":"Electronic Journal of Probability","publication_identifier":{"issn":["1083-6489"]},"department":[{"_id":"LaEr"}],"ec_funded":1,"ddc":["510"],"doi":"10.1214/EJP.v19-3054","has_accepted_license":"1","language":[{"iso":"eng"}],"volume":19,"file":[{"date_created":"2018-12-12T10:14:06Z","access_level":"open_access","content_type":"application/pdf","file_id":"5055","creator":"system","relation":"main_file","file_size":810150,"checksum":"7eb297ff367a2ee73b21b6dd1e1948e4","date_updated":"2020-07-14T12:45:34Z","file_name":"IST-2016-427-v1+1_3054-16624-4-PB.pdf"}],"project":[{"grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems"}],"fulldoi":"https://doi.org/10.1214/EJP.v19-3054","oa_version":"Published Version","citation":{"mla":"Bloemendal, Alex, et al. “Isotropic Local Laws for Sample Covariance and Generalized Wigner Matrices.” <i>Electronic Journal of Probability</i>, vol. 19, 33, Institute of Mathematical Statistics, 2014, doi:<a href=\"https://doi.org/10.1214/EJP.v19-3054\">10.1214/EJP.v19-3054</a>.","apa":"Bloemendal, A., Erdös, L., Knowles, A., Yau, H., &#38; Yin, J. (2014). Isotropic local laws for sample covariance and generalized Wigner matrices. <i>Electronic Journal of Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/EJP.v19-3054\">https://doi.org/10.1214/EJP.v19-3054</a>","short":"A. Bloemendal, L. Erdös, A. Knowles, H. Yau, J. Yin, Electronic Journal of Probability 19 (2014).","chicago":"Bloemendal, Alex, László Erdös, Antti Knowles, Horng Yau, and Jun Yin. “Isotropic Local Laws for Sample Covariance and Generalized Wigner Matrices.” <i>Electronic Journal of Probability</i>. Institute of Mathematical Statistics, 2014. <a href=\"https://doi.org/10.1214/EJP.v19-3054\">https://doi.org/10.1214/EJP.v19-3054</a>.","ama":"Bloemendal A, Erdös L, Knowles A, Yau H, Yin J. Isotropic local laws for sample covariance and generalized Wigner matrices. <i>Electronic Journal of Probability</i>. 2014;19. doi:<a href=\"https://doi.org/10.1214/EJP.v19-3054\">10.1214/EJP.v19-3054</a>","ista":"Bloemendal A, Erdös L, Knowles A, Yau H, Yin J. 2014. Isotropic local laws for sample covariance and generalized Wigner matrices. Electronic Journal of Probability. 19, 33.","ieee":"A. Bloemendal, L. Erdös, A. Knowles, H. Yau, and J. Yin, “Isotropic local laws for sample covariance and generalized Wigner matrices,” <i>Electronic Journal of Probability</i>, vol. 19. Institute of Mathematical Statistics, 2014."},"scopus_import":"1","publisher":"Institute of Mathematical Statistics","year":"2014"},{"status":"public","article_processing_charge":"No","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_created":"2018-12-11T11:59:08Z","_id":"2699","page":"1127 - 1190","author":[{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","last_name":"Erdös","full_name":"Erdös, László","orcid":"0000-0001-5366-9603"},{"first_name":"Paul","last_name":"Bourgade","full_name":"Bourgade, Paul"},{"full_name":"Yau, Horng","last_name":"Yau","first_name":"Horng"}],"type":"journal_article","isi":1,"issue":"6","abstract":[{"text":"We prove the universality of the β-ensembles with convex analytic potentials and for any β &gt;\r\n0, i.e. we show that the spacing distributions of log-gases at any inverse temperature β coincide with those of the Gaussian β-ensembles.","lang":"eng"}],"intvolume":"       163","publist_id":"4197","month":"04","date_updated":"2025-09-29T11:11:17Z","date_published":"2014-04-01T00:00:00Z","oa":1,"quality_controlled":"1","day":"01","fulldoi":"https://doi.org/10.1215/00127094-2649752","oa_version":"Preprint","citation":{"ieee":"L. Erdös, P. Bourgade, and H. Yau, “Universality of general β-ensembles,” <i>Duke Mathematical Journal</i>, vol. 163, no. 6. Duke University Press, pp. 1127–1190, 2014.","ista":"Erdös L, Bourgade P, Yau H. 2014. Universality of general β-ensembles. Duke Mathematical Journal. 163(6), 1127–1190.","chicago":"Erdös, László, Paul Bourgade, and Horng Yau. “Universality of General β-Ensembles.” <i>Duke Mathematical Journal</i>. Duke University Press, 2014. <a href=\"https://doi.org/10.1215/00127094-2649752\">https://doi.org/10.1215/00127094-2649752</a>.","ama":"Erdös L, Bourgade P, Yau H. Universality of general β-ensembles. <i>Duke Mathematical Journal</i>. 2014;163(6):1127-1190. doi:<a href=\"https://doi.org/10.1215/00127094-2649752\">10.1215/00127094-2649752</a>","apa":"Erdös, L., Bourgade, P., &#38; Yau, H. (2014). Universality of general β-ensembles. <i>Duke Mathematical Journal</i>. Duke University Press. <a href=\"https://doi.org/10.1215/00127094-2649752\">https://doi.org/10.1215/00127094-2649752</a>","mla":"Erdös, László, et al. “Universality of General β-Ensembles.” <i>Duke Mathematical Journal</i>, vol. 163, no. 6, Duke University Press, 2014, pp. 1127–90, doi:<a href=\"https://doi.org/10.1215/00127094-2649752\">10.1215/00127094-2649752</a>.","short":"L. Erdös, P. Bourgade, H. Yau, Duke Mathematical Journal 163 (2014) 1127–1190."},"scopus_import":"1","language":[{"iso":"eng"}],"volume":163,"main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1104.2272"}],"publisher":"Duke University Press","year":"2014","publication":"Duke Mathematical Journal","title":"Universality of general β-ensembles","publication_status":"published","external_id":{"arxiv":["1104.2272"],"isi":["000334478100003"]},"doi":"10.1215/00127094-2649752","department":[{"_id":"LaEr"}],"arxiv":1},{"publist_id":"4803","intvolume":"        19","abstract":[{"text":"We extend the proof of the local semicircle law for generalized Wigner matrices given in MR3068390 to the case when the matrix of variances has an eigenvalue -1. In particular, this result provides a short proof of the optimal local Marchenko-Pastur law at the hard edge (i.e. around zero) for sample covariance matrices X*X, where the variances of the entries of X may vary.","lang":"eng"}],"file_date_updated":"2020-07-14T12:45:31Z","day":"09","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"quality_controlled":"1","oa":1,"date_published":"2014-06-09T00:00:00Z","date_updated":"2025-09-29T11:35:54Z","month":"06","_id":"2179","pubrep_id":"426","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_created":"2018-12-11T11:56:10Z","article_processing_charge":"No","status":"public","type":"journal_article","isi":1,"author":[{"id":"36F2FB7E-F248-11E8-B48F-1D18A9856A87","first_name":"Oskari H","full_name":"Ajanki, Oskari H","last_name":"Ajanki"},{"last_name":"Erdös","orcid":"0000-0001-5366-9603","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László"},{"id":"3020C786-F248-11E8-B48F-1D18A9856A87","first_name":"Torben H","last_name":"Krüger","orcid":"0000-0002-4821-3297","full_name":"Krüger, Torben H"}],"corr_author":"1","external_id":{"isi":["000341869300001"]},"publication_status":"published","title":"Local semicircle law with imprimitive variance matrix","publication":"Electronic Communications in Probability","department":[{"_id":"LaEr"}],"has_accepted_license":"1","doi":"10.1214/ECP.v19-3121","ddc":["570"],"volume":19,"language":[{"iso":"eng"}],"oa_version":"Published Version","scopus_import":"1","citation":{"short":"O.H. Ajanki, L. Erdös, T.H. Krüger, Electronic Communications in Probability 19 (2014).","apa":"Ajanki, O. H., Erdös, L., &#38; Krüger, T. H. (2014). Local semicircle law with imprimitive variance matrix. <i>Electronic Communications in Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/ECP.v19-3121\">https://doi.org/10.1214/ECP.v19-3121</a>","mla":"Ajanki, Oskari H., et al. “Local Semicircle Law with Imprimitive Variance Matrix.” <i>Electronic Communications in Probability</i>, vol. 19, Institute of Mathematical Statistics, 2014, doi:<a href=\"https://doi.org/10.1214/ECP.v19-3121\">10.1214/ECP.v19-3121</a>.","ieee":"O. H. Ajanki, L. Erdös, and T. H. Krüger, “Local semicircle law with imprimitive variance matrix,” <i>Electronic Communications in Probability</i>, vol. 19. Institute of Mathematical Statistics, 2014.","ista":"Ajanki OH, Erdös L, Krüger TH. 2014. Local semicircle law with imprimitive variance matrix. Electronic Communications in Probability. 19.","ama":"Ajanki OH, Erdös L, Krüger TH. Local semicircle law with imprimitive variance matrix. <i>Electronic Communications in Probability</i>. 2014;19. doi:<a href=\"https://doi.org/10.1214/ECP.v19-3121\">10.1214/ECP.v19-3121</a>","chicago":"Ajanki, Oskari H, László Erdös, and Torben H Krüger. “Local Semicircle Law with Imprimitive Variance Matrix.” <i>Electronic Communications in Probability</i>. Institute of Mathematical Statistics, 2014. <a href=\"https://doi.org/10.1214/ECP.v19-3121\">https://doi.org/10.1214/ECP.v19-3121</a>."},"fulldoi":"https://doi.org/10.1214/ECP.v19-3121","file":[{"date_created":"2018-12-12T10:09:06Z","access_level":"open_access","creator":"system","relation":"main_file","file_id":"4729","content_type":"application/pdf","file_name":"IST-2016-426-v1+1_3121-17518-1-PB.pdf","date_updated":"2020-07-14T12:45:31Z","file_size":327322,"checksum":"bd8a041c76d62fe820bf73ff13ce7d1b"}],"year":"2014","publisher":"Institute of Mathematical Statistics"},{"author":[{"orcid":"0000-0001-5366-9603","full_name":"Erdös, László","last_name":"Erdös","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"}],"page":"214 - 236","conference":{"location":"Seoul, Korea","name":"ICM: International Congress of Mathematicians","end_date":"2014-08-21","start_date":"2014-08-13"},"type":"conference","article_processing_charge":"No","status":"public","_id":"1507","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2018-12-11T11:52:25Z","date_updated":"2025-06-11T07:52:59Z","month":"08","oa":1,"day":"01","quality_controlled":"1","date_published":"2014-08-01T00:00:00Z","abstract":[{"text":"The Wigner-Dyson-Gaudin-Mehta conjecture asserts that the local eigenvalue statistics of large real and complex Hermitian matrices with independent, identically distributed entries are universal in a sense that they depend only on the symmetry class of the matrix and otherwise are independent of the details of the distribution. We present the recent solution to this half-century old conjecture. We explain how stochastic tools, such as the Dyson Brownian motion, and PDE ideas, such as De Giorgi-Nash-Moser regularity theory, were combined in the solution. We also show related results for log-gases that represent a universal model for strongly correlated systems. Finally, in the spirit of Wigner’s original vision, we discuss the extensions of these universality results to more realistic physical systems such as random band matrices.","lang":"eng"}],"intvolume":"         3","acknowledgement":"The author is partially supported by SFB-TR 12 Grant of the German Research Council.","publist_id":"5670","publisher":"International Congress of Mathematicians","year":"2014","citation":{"chicago":"Erdös, László. “Random Matrices, Log-Gases and Hölder Regularity.” In <i>Proceedings of the International Congress of Mathematicians</i>, 3:214–36. International Congress of Mathematicians, 2014.","ama":"Erdös L. Random matrices, log-gases and Hölder regularity. In: <i>Proceedings of the International Congress of Mathematicians</i>. Vol 3. International Congress of Mathematicians; 2014:214-236.","ieee":"L. Erdös, “Random matrices, log-gases and Hölder regularity,” in <i>Proceedings of the International Congress of Mathematicians</i>, Seoul, Korea, 2014, vol. 3, pp. 214–236.","ista":"Erdös L. 2014. Random matrices, log-gases and Hölder regularity. Proceedings of the International Congress of Mathematicians. ICM: International Congress of Mathematicians vol. 3, 214–236.","short":"L. Erdös, in:, Proceedings of the International Congress of Mathematicians, International Congress of Mathematicians, 2014, pp. 214–236.","mla":"Erdös, László. “Random Matrices, Log-Gases and Hölder Regularity.” <i>Proceedings of the International Congress of Mathematicians</i>, vol. 3, International Congress of Mathematicians, 2014, pp. 214–36.","apa":"Erdös, L. (2014). Random matrices, log-gases and Hölder regularity. In <i>Proceedings of the International Congress of Mathematicians</i> (Vol. 3, pp. 214–236). Seoul, Korea: International Congress of Mathematicians."},"oa_version":"Submitted Version","scopus_import":"1","project":[{"grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems"}],"volume":3,"main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1407.5752"}],"language":[{"iso":"eng"}],"arxiv":1,"ec_funded":1,"department":[{"_id":"LaEr"}],"publication":"Proceedings of the International Congress of Mathematicians","external_id":{"arxiv":["1407.5752"]},"title":"Random matrices, log-gases and Hölder regularity","publication_status":"published"},{"month":"12","date_updated":"2025-09-29T12:14:12Z","date_published":"2014-12-17T00:00:00Z","day":"17","quality_controlled":"1","oa":1,"intvolume":"        17","abstract":[{"lang":"eng","text":"We consider cross products of finite graphs with a class of trees that have arbitrarily but finitely long line segments, such as the Fibonacci tree. Such cross products are called tree-strips. We prove that for small disorder random Schrödinger operators on such tree-strips have purely absolutely continuous spectrum in a certain set."}],"publist_id":"5168","corr_author":"1","article_type":"original","page":"409 - 440","author":[{"id":"4760E9F8-F248-11E8-B48F-1D18A9856A87","first_name":"Christian","orcid":"0000-0001-8255-3968","full_name":"Sadel, Christian","last_name":"Sadel"}],"type":"journal_article","isi":1,"issue":"3-4","status":"public","article_processing_charge":"No","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_created":"2018-12-11T11:54:45Z","_id":"1926","doi":"10.1007/s11040-014-9163-4","department":[{"_id":"LaEr"}],"ec_funded":1,"arxiv":1,"publication":"Mathematical Physics, Analysis and Geometry","publication_status":"published","title":"Absolutely continuous spectrum for random Schrödinger operators on the Fibonacci and similar Tree-strips","external_id":{"arxiv":["1304.3862"],"isi":["000348286700010"]},"year":"2014","publisher":"Springer","fulldoi":"https://doi.org/10.1007/s11040-014-9163-4","project":[{"name":"NSERC Postdoctoral fellowship","_id":"26450934-B435-11E9-9278-68D0E5697425"},{"call_identifier":"FP7","name":"International IST Postdoc Fellowship Programme","grant_number":"291734","_id":"25681D80-B435-11E9-9278-68D0E5697425"}],"oa_version":"Preprint","scopus_import":"1","citation":{"mla":"Sadel, Christian. “Absolutely Continuous Spectrum for Random Schrödinger Operators on the Fibonacci and Similar Tree-Strips.” <i>Mathematical Physics, Analysis and Geometry</i>, vol. 17, no. 3–4, Springer, 2014, pp. 409–40, doi:<a href=\"https://doi.org/10.1007/s11040-014-9163-4\">10.1007/s11040-014-9163-4</a>.","apa":"Sadel, C. (2014). Absolutely continuous spectrum for random Schrödinger operators on the Fibonacci and similar Tree-strips. <i>Mathematical Physics, Analysis and Geometry</i>. Springer. <a href=\"https://doi.org/10.1007/s11040-014-9163-4\">https://doi.org/10.1007/s11040-014-9163-4</a>","short":"C. Sadel, Mathematical Physics, Analysis and Geometry 17 (2014) 409–440.","ama":"Sadel C. Absolutely continuous spectrum for random Schrödinger operators on the Fibonacci and similar Tree-strips. <i>Mathematical Physics, Analysis and Geometry</i>. 2014;17(3-4):409-440. doi:<a href=\"https://doi.org/10.1007/s11040-014-9163-4\">10.1007/s11040-014-9163-4</a>","chicago":"Sadel, Christian. “Absolutely Continuous Spectrum for Random Schrödinger Operators on the Fibonacci and Similar Tree-Strips.” <i>Mathematical Physics, Analysis and Geometry</i>. Springer, 2014. <a href=\"https://doi.org/10.1007/s11040-014-9163-4\">https://doi.org/10.1007/s11040-014-9163-4</a>.","ieee":"C. Sadel, “Absolutely continuous spectrum for random Schrödinger operators on the Fibonacci and similar Tree-strips,” <i>Mathematical Physics, Analysis and Geometry</i>, vol. 17, no. 3–4. Springer, pp. 409–440, 2014.","ista":"Sadel C. 2014. Absolutely continuous spectrum for random Schrödinger operators on the Fibonacci and similar Tree-strips. Mathematical Physics, Analysis and Geometry. 17(3–4), 409–440."},"language":[{"iso":"eng"}],"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1304.3862"}],"volume":17},{"month":"11","date_updated":"2025-09-29T12:08:21Z","date_published":"2014-11-01T00:00:00Z","oa":1,"day":"01","quality_controlled":"1","abstract":[{"text":"We prove the edge universality of the beta ensembles for any β ≥ 1, provided that the limiting spectrum is supported on a single interval, and the external potential is C4 and regular. We also prove that the edge universality holds for generalized Wigner matrices for all symmetry classes. Moreover, our results allow us to extend bulk universality for beta ensembles from analytic potentials to potentials in class C4.","lang":"eng"}],"intvolume":"       332","publist_id":"5158","page":"261 - 353","author":[{"full_name":"Bourgade, Paul","last_name":"Bourgade","first_name":"Paul"},{"orcid":"0000-0001-5366-9603","full_name":"Erdös, László","last_name":"Erdös","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László"},{"first_name":"Horngtzer","last_name":"Yau","full_name":"Yau, Horngtzer"}],"isi":1,"type":"journal_article","issue":"1","status":"public","article_processing_charge":"No","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_created":"2018-12-11T11:54:48Z","_id":"1937","doi":"10.1007/s00220-014-2120-z","department":[{"_id":"LaEr"}],"arxiv":1,"publication":"Communications in Mathematical Physics","title":"Edge universality of beta ensembles","publication_status":"published","external_id":{"arxiv":["1306.5728"],"isi":["000341491700007"]},"publisher":"Springer","year":"2014","project":[{"name":"Glutamaterge synaptische Übertragung und Plastizität in hippocampalen Mikroschaltkreisen","grant_number":"SFB-TR3-TP10B","_id":"25BDE9A4-B435-11E9-9278-68D0E5697425"}],"fulldoi":"https://doi.org/10.1007/s00220-014-2120-z","scopus_import":"1","oa_version":"Submitted Version","citation":{"mla":"Bourgade, Paul, et al. “Edge Universality of Beta Ensembles.” <i>Communications in Mathematical Physics</i>, vol. 332, no. 1, Springer, 2014, pp. 261–353, doi:<a href=\"https://doi.org/10.1007/s00220-014-2120-z\">10.1007/s00220-014-2120-z</a>.","apa":"Bourgade, P., Erdös, L., &#38; Yau, H. (2014). Edge universality of beta ensembles. <i>Communications in Mathematical Physics</i>. Springer. <a href=\"https://doi.org/10.1007/s00220-014-2120-z\">https://doi.org/10.1007/s00220-014-2120-z</a>","short":"P. Bourgade, L. Erdös, H. Yau, Communications in Mathematical Physics 332 (2014) 261–353.","chicago":"Bourgade, Paul, László Erdös, and Horngtzer Yau. “Edge Universality of Beta Ensembles.” <i>Communications in Mathematical Physics</i>. Springer, 2014. <a href=\"https://doi.org/10.1007/s00220-014-2120-z\">https://doi.org/10.1007/s00220-014-2120-z</a>.","ama":"Bourgade P, Erdös L, Yau H. Edge universality of beta ensembles. <i>Communications in Mathematical Physics</i>. 2014;332(1):261-353. doi:<a href=\"https://doi.org/10.1007/s00220-014-2120-z\">10.1007/s00220-014-2120-z</a>","ieee":"P. Bourgade, L. Erdös, and H. Yau, “Edge universality of beta ensembles,” <i>Communications in Mathematical Physics</i>, vol. 332, no. 1. Springer, pp. 261–353, 2014.","ista":"Bourgade P, Erdös L, Yau H. 2014. Edge universality of beta ensembles. Communications in Mathematical Physics. 332(1), 261–353."},"language":[{"iso":"eng"}],"volume":332,"main_file_link":[{"url":"http://arxiv.org/abs/1306.5728","open_access":"1"}]},{"language":[{"iso":"eng"}],"main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1407.1552"}],"volume":17,"project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804","name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7"}],"fulldoi":"https://doi.org/10.1007/s11040-014-9164-3","oa_version":"Submitted Version","scopus_import":"1","citation":{"ista":"Erdös L, Schröder DJ. 2014. Phase transition in the density of states of quantum spin glasses. Mathematical Physics, Analysis and Geometry. 17(3–4), 441–464.","ieee":"L. Erdös and D. J. Schröder, “Phase transition in the density of states of quantum spin glasses,” <i>Mathematical Physics, Analysis and Geometry</i>, vol. 17, no. 3–4. Springer, pp. 441–464, 2014.","ama":"Erdös L, Schröder DJ. Phase transition in the density of states of quantum spin glasses. <i>Mathematical Physics, Analysis and Geometry</i>. 2014;17(3-4):441-464. doi:<a href=\"https://doi.org/10.1007/s11040-014-9164-3\">10.1007/s11040-014-9164-3</a>","chicago":"Erdös, László, and Dominik J Schröder. “Phase Transition in the Density of States of Quantum Spin Glasses.” <i>Mathematical Physics, Analysis and Geometry</i>. Springer, 2014. <a href=\"https://doi.org/10.1007/s11040-014-9164-3\">https://doi.org/10.1007/s11040-014-9164-3</a>.","apa":"Erdös, L., &#38; Schröder, D. J. (2014). Phase transition in the density of states of quantum spin glasses. <i>Mathematical Physics, Analysis and Geometry</i>. Springer. <a href=\"https://doi.org/10.1007/s11040-014-9164-3\">https://doi.org/10.1007/s11040-014-9164-3</a>","mla":"Erdös, László, and Dominik J. Schröder. “Phase Transition in the Density of States of Quantum Spin Glasses.” <i>Mathematical Physics, Analysis and Geometry</i>, vol. 17, no. 3–4, Springer, 2014, pp. 441–64, doi:<a href=\"https://doi.org/10.1007/s11040-014-9164-3\">10.1007/s11040-014-9164-3</a>.","short":"L. Erdös, D.J. Schröder, Mathematical Physics, Analysis and Geometry 17 (2014) 441–464."},"year":"2014","publisher":"Springer","publication_status":"published","title":"Phase transition in the density of states of quantum spin glasses","external_id":{"isi":["000348286700011"],"arxiv":["1407.1552"]},"publication":"Mathematical Physics, Analysis and Geometry","department":[{"_id":"LaEr"}],"ec_funded":1,"arxiv":1,"doi":"10.1007/s11040-014-9164-3","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_created":"2018-12-11T11:55:15Z","_id":"2019","status":"public","article_processing_charge":"No","issue":"3-4","type":"journal_article","isi":1,"page":"441 - 464","author":[{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","orcid":"0000-0001-5366-9603","full_name":"Erdös, László","last_name":"Erdös"},{"full_name":"Schröder, Dominik J","last_name":"Schröder","first_name":"Dominik J"}],"publist_id":"5053","intvolume":"        17","abstract":[{"lang":"eng","text":"We prove that the empirical density of states of quantum spin glasses on arbitrary graphs converges to a normal distribution as long as the maximal degree is negligible compared with the total number of edges. This extends the recent results of Keating et al. (2014) that were proved for graphs with bounded chromatic number and with symmetric coupling distribution. Furthermore, we generalise the result to arbitrary hypergraphs. We test the optimality of our condition on the maximal degree for p-uniform hypergraphs that correspond to p-spin glass Hamiltonians acting on n distinguishable spin- 1/2 particles. At the critical threshold p = n1/2 we find a sharp classical-quantum phase transition between the normal distribution and the Wigner semicircle law. The former is characteristic to classical systems with commuting variables, while the latter is a signature of noncommutative random matrix theory."}],"date_published":"2014-12-17T00:00:00Z","day":"17","quality_controlled":"1","oa":1,"month":"12","date_updated":"2025-09-29T12:00:05Z"},{"status":"public","article_processing_charge":"No","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_created":"2018-12-11T11:59:34Z","_id":"2782","corr_author":"1","page":"1003 - 1032","author":[{"last_name":"Erdös","orcid":"0000-0001-5366-9603","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László"},{"first_name":"Brendan","last_name":"Farrell","full_name":"Farrell, Brendan"}],"type":"journal_article","isi":1,"issue":"6","abstract":[{"lang":"eng","text":"We consider random n×n matrices of the form (XX*+YY*)^{-1/2}YY*(XX*+YY*)^{-1/2}, where X and Y have independent entries with zero mean and variance one. These matrices are the natural generalization of the Gaussian case, which are known as MANOVA matrices and which have joint eigenvalue density given by the third classical ensemble, the Jacobi ensemble. We show that, away from the spectral edge, the eigenvalue density converges to the limiting density of the Jacobi ensemble even on the shortest possible scales of order 1/n (up to log n factors). This result is the analogue of the local Wigner semicircle law and the local Marchenko-Pastur law for general MANOVA matrices."}],"intvolume":"       152","publist_id":"4107","month":"07","date_updated":"2025-09-29T14:07:43Z","date_published":"2013-07-18T00:00:00Z","oa":1,"day":"18","quality_controlled":"1","fulldoi":"https://doi.org/10.1007/s10955-013-0807-8","scopus_import":"1","oa_version":"Preprint","citation":{"apa":"Erdös, L., &#38; Farrell, B. (2013). Local eigenvalue density for general MANOVA matrices. <i>Journal of Statistical Physics</i>. Springer. <a href=\"https://doi.org/10.1007/s10955-013-0807-8\">https://doi.org/10.1007/s10955-013-0807-8</a>","mla":"Erdös, László, and Brendan Farrell. “Local Eigenvalue Density for General MANOVA Matrices.” <i>Journal of Statistical Physics</i>, vol. 152, no. 6, Springer, 2013, pp. 1003–32, doi:<a href=\"https://doi.org/10.1007/s10955-013-0807-8\">10.1007/s10955-013-0807-8</a>.","short":"L. Erdös, B. Farrell, Journal of Statistical Physics 152 (2013) 1003–1032.","ista":"Erdös L, Farrell B. 2013. Local eigenvalue density for general MANOVA matrices. Journal of Statistical Physics. 152(6), 1003–1032.","ieee":"L. Erdös and B. Farrell, “Local eigenvalue density for general MANOVA matrices,” <i>Journal of Statistical Physics</i>, vol. 152, no. 6. Springer, pp. 1003–1032, 2013.","chicago":"Erdös, László, and Brendan Farrell. “Local Eigenvalue Density for General MANOVA Matrices.” <i>Journal of Statistical Physics</i>. Springer, 2013. <a href=\"https://doi.org/10.1007/s10955-013-0807-8\">https://doi.org/10.1007/s10955-013-0807-8</a>.","ama":"Erdös L, Farrell B. Local eigenvalue density for general MANOVA matrices. <i>Journal of Statistical Physics</i>. 2013;152(6):1003-1032. doi:<a href=\"https://doi.org/10.1007/s10955-013-0807-8\">10.1007/s10955-013-0807-8</a>"},"language":[{"iso":"eng"}],"volume":152,"main_file_link":[{"url":"http://arxiv.org/abs/1207.0031","open_access":"1"}],"publisher":"Springer","year":"2013","publication":"Journal of Statistical Physics","title":"Local eigenvalue density for general MANOVA matrices","publication_status":"published","external_id":{"arxiv":["1207.0031"],"isi":["000323203800001"]},"doi":"10.1007/s10955-013-0807-8","department":[{"_id":"LaEr"}],"arxiv":1},{"abstract":[{"lang":"eng","text":"We consider a general class of N × N random matrices whose entries hij are independent up to a symmetry constraint, but not necessarily identically distributed. Our main result is a local semicircle law which improves previous results [17] both in the bulk and at the edge. The error bounds are given in terms of the basic small parameter of the model, maxi,j E|hij|2. As a consequence, we prove the universality of the local n-point correlation functions in the bulk spectrum for a class of matrices whose entries do not have comparable variances, including random band matrices with band width W ≫N1-εn with some εn &gt; 0 and with a negligible mean-field component. In addition, we provide a coherent and pedagogical proof of the local semicircle law, streamlining and strengthening previous arguments from [17, 19, 6]."}],"file_date_updated":"2020-07-14T12:45:50Z","intvolume":"        18","publist_id":"3962","date_updated":"2025-09-29T13:46:52Z","month":"05","oa":1,"quality_controlled":"1","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"day":"29","date_published":"2013-05-29T00:00:00Z","article_processing_charge":"No","status":"public","_id":"2837","date_created":"2018-12-11T11:59:51Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","pubrep_id":"406","author":[{"orcid":"0000-0001-5366-9603","full_name":"Erdös, László","last_name":"Erdös","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László"},{"full_name":"Knowles, Antti","last_name":"Knowles","first_name":"Antti"},{"last_name":"Yau","full_name":"Yau, Horng","first_name":"Horng"},{"first_name":"Jun","last_name":"Yin","full_name":"Yin, Jun"}],"page":"1-58","isi":1,"issue":"59","type":"journal_article","publication":"Electronic Journal of Probability","external_id":{"isi":["000319561600001"]},"title":"The local semicircle law for a general class of random matrices","publication_status":"published","has_accepted_license":"1","ddc":["530"],"doi":"10.1214/EJP.v18-2473","department":[{"_id":"LaEr"}],"oa_version":"Published Version","scopus_import":"1","citation":{"apa":"Erdös, L., Knowles, A., Yau, H., &#38; Yin, J. (2013). The local semicircle law for a general class of random matrices. <i>Electronic Journal of Probability</i>. Institute of Mathematical Statistics. <a href=\"https://doi.org/10.1214/EJP.v18-2473\">https://doi.org/10.1214/EJP.v18-2473</a>","mla":"Erdös, László, et al. “The Local Semicircle Law for a General Class of Random Matrices.” <i>Electronic Journal of Probability</i>, vol. 18, no. 59, Institute of Mathematical Statistics, 2013, pp. 1–58, doi:<a href=\"https://doi.org/10.1214/EJP.v18-2473\">10.1214/EJP.v18-2473</a>.","short":"L. Erdös, A. Knowles, H. Yau, J. Yin, Electronic Journal of Probability 18 (2013) 1–58.","ieee":"L. Erdös, A. Knowles, H. Yau, and J. Yin, “The local semicircle law for a general class of random matrices,” <i>Electronic Journal of Probability</i>, vol. 18, no. 59. Institute of Mathematical Statistics, pp. 1–58, 2013.","ista":"Erdös L, Knowles A, Yau H, Yin J. 2013. The local semicircle law for a general class of random matrices. Electronic Journal of Probability. 18(59), 1–58.","ama":"Erdös L, Knowles A, Yau H, Yin J. The local semicircle law for a general class of random matrices. <i>Electronic Journal of Probability</i>. 2013;18(59):1-58. doi:<a href=\"https://doi.org/10.1214/EJP.v18-2473\">10.1214/EJP.v18-2473</a>","chicago":"Erdös, László, Antti Knowles, Horng Yau, and Jun Yin. “The Local Semicircle Law for a General Class of Random Matrices.” <i>Electronic Journal of Probability</i>. Institute of Mathematical Statistics, 2013. <a href=\"https://doi.org/10.1214/EJP.v18-2473\">https://doi.org/10.1214/EJP.v18-2473</a>."},"file":[{"access_level":"open_access","date_created":"2018-12-12T10:15:46Z","file_name":"IST-2016-406-v1+1_2473-13759-1-PB.pdf","date_updated":"2020-07-14T12:45:50Z","checksum":"aac9e52a00cb2f5149dc9e362b5ccf44","file_size":651497,"file_id":"5169","content_type":"application/pdf","relation":"main_file","creator":"system"}],"fulldoi":"https://doi.org/10.1214/EJP.v18-2473","volume":18,"language":[{"iso":"eng"}],"publisher":"Institute of Mathematical Statistics","year":"2013"},{"language":[{"iso":"eng"}],"volume":15,"main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1105.0506"}],"fulldoi":"https://doi.org/10.4171/JEMS/416","oa_version":"Preprint","citation":{"ieee":"L. Erdös, S. Fournais, and J. Solovej, “Stability and semiclassics in self-generated fields,” <i>Journal of the European Mathematical Society</i>, vol. 15, no. 6. EMS Press, pp. 2093–2113, 2013.","ista":"Erdös L, Fournais S, Solovej J. 2013. Stability and semiclassics in self-generated fields. Journal of the European Mathematical Society. 15(6), 2093–2113.","chicago":"Erdös, László, Søren Fournais, and Jan Solovej. “Stability and Semiclassics in Self-Generated Fields.” <i>Journal of the European Mathematical Society</i>. EMS Press, 2013. <a href=\"https://doi.org/10.4171/JEMS/416\">https://doi.org/10.4171/JEMS/416</a>.","ama":"Erdös L, Fournais S, Solovej J. Stability and semiclassics in self-generated fields. <i>Journal of the European Mathematical Society</i>. 2013;15(6):2093-2113. doi:<a href=\"https://doi.org/10.4171/JEMS/416\">10.4171/JEMS/416</a>","apa":"Erdös, L., Fournais, S., &#38; Solovej, J. (2013). Stability and semiclassics in self-generated fields. <i>Journal of the European Mathematical Society</i>. EMS Press. <a href=\"https://doi.org/10.4171/JEMS/416\">https://doi.org/10.4171/JEMS/416</a>","mla":"Erdös, László, et al. “Stability and Semiclassics in Self-Generated Fields.” <i>Journal of the European Mathematical Society</i>, vol. 15, no. 6, EMS Press, 2013, pp. 2093–113, doi:<a href=\"https://doi.org/10.4171/JEMS/416\">10.4171/JEMS/416</a>.","short":"L. Erdös, S. Fournais, J. Solovej, Journal of the European Mathematical Society 15 (2013) 2093–2113."},"publisher":"EMS Press","year":"2013","title":"Stability and semiclassics in self-generated fields","publication_status":"published","external_id":{"isi":["000326323400006"],"arxiv":["1105.0506"]},"publication":"Journal of the European Mathematical Society","department":[{"_id":"LaEr"}],"arxiv":1,"doi":"10.4171/JEMS/416","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2018-12-11T11:59:07Z","_id":"2698","status":"public","article_processing_charge":"No","isi":1,"issue":"6","type":"journal_article","page":"2093 - 2113","author":[{"last_name":"Erdös","full_name":"Erdös, László","orcid":"0000-0001-5366-9603","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Søren","last_name":"Fournais","full_name":"Fournais, Søren"},{"first_name":"Jan","last_name":"Solovej","full_name":"Solovej, Jan"}],"publist_id":"4198","abstract":[{"lang":"eng","text":"We consider non-interacting particles subject to a fixed external potential V and a self-generated magnetic field B. The total energy includes the field energy β∫B2 and we minimize over all particle states and magnetic fields. In the case of spin-1/2 particles this minimization leads to the coupled Maxwell-Pauli system. The parameter β tunes the coupling strength between the field and the particles and it effectively determines the strength of the field. We investigate the stability and the semiclassical asymptotics, h→0, of the total ground state energy E(β,h,V). The relevant parameter measuring the field strength in the semiclassical limit is κ=βh. We are not able to give the exact leading order semiclassical asymptotics uniformly in κ or even for fixed κ. We do however give upper and lower bounds on E with almost matching dependence on κ. In the simultaneous limit h→0 and κ→∞ we show that the standard non-magnetic Weyl asymptotics holds. The same result also holds for the spinless case, i.e. where the Pauli operator is replaced by the Schrödinger operator."}],"intvolume":"        15","date_published":"2013-10-16T00:00:00Z","oa":1,"quality_controlled":"1","day":"16","month":"10","date_updated":"2026-07-06T11:56:35Z"}]