[{"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","citation":{"ieee":"Z. Bao, G. Pan, and W. Zhou, “Universality for the largest eigenvalue of sample covariance matrices with general population,” Annals of Statistics, vol. 43, no. 1. Institute of Mathematical Statistics, pp. 382–421, 2015.","ista":"Bao Z, Pan G, Zhou W. 2015. Universality for the largest eigenvalue of sample covariance matrices with general population. Annals of Statistics. 43(1), 382–421.","apa":"Bao, Z., Pan, G., & Zhou, W. (2015). Universality for the largest eigenvalue of sample covariance matrices with general population. Annals of Statistics. Institute of Mathematical Statistics. https://doi.org/10.1214/14-AOS1281","mla":"Bao, Zhigang, et al. “Universality for the Largest Eigenvalue of Sample Covariance Matrices with General Population.” Annals of Statistics, vol. 43, no. 1, Institute of Mathematical Statistics, 2015, pp. 382–421, doi:10.1214/14-AOS1281.","short":"Z. Bao, G. Pan, W. Zhou, Annals of Statistics 43 (2015) 382–421.","chicago":"Bao, Zhigang, Guangming Pan, and Wang Zhou. “Universality for the Largest Eigenvalue of Sample Covariance Matrices with General Population.” Annals of Statistics. Institute of Mathematical Statistics, 2015. https://doi.org/10.1214/14-AOS1281.","ama":"Bao Z, Pan G, Zhou W. Universality for the largest eigenvalue of sample covariance matrices with general population. Annals of Statistics. 2015;43(1):382-421. doi:10.1214/14-AOS1281"},"arxiv":1,"oa":1,"date_published":"2015-02-01T00:00:00Z","_id":"1505","author":[{"last_name":"Bao","orcid":"0000-0003-3036-1475","first_name":"Zhigang","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","full_name":"Bao, Zhigang"},{"full_name":"Pan, Guangming","first_name":"Guangming","last_name":"Pan"},{"last_name":"Zhou","first_name":"Wang","full_name":"Zhou, Wang"}],"date_created":"2018-12-11T11:52:25Z","doi":"10.1214/14-AOS1281","volume":43,"acknowledgement":"B.Z. was supported in part by NSFC Grant 11071213, ZJNSF Grant R6090034 and SRFDP Grant 20100101110001. P.G. was supported in part by the Ministry of Education, Singapore, under Grant ARC 14/11. Z.W. was supported in part by the Ministry of Education, Singapore, under Grant ARC 14/11, and by a Grant R-155-000-131-112 at the National University of Singapore\r\n","intvolume":" 43","quality_controlled":"1","main_file_link":[{"url":"https://arxiv.org/abs/1304.5690","open_access":"1"}],"page":"382 - 421","title":"Universality for the largest eigenvalue of sample covariance matrices with general population","oa_version":"Preprint","publication_status":"published","abstract":[{"lang":"eng","text":"This paper is aimed at deriving the universality of the largest eigenvalue of a class of high-dimensional real or complex sample covariance matrices of the form W N =Σ 1/2XX∗Σ 1/2 . Here, X = (xij )M,N is an M× N random matrix with independent entries xij , 1 ≤ i M,≤ 1 ≤ j ≤ N such that Exij = 0, E|xij |2 = 1/N . On dimensionality, we assume that M = M(N) and N/M → d ε (0, ∞) as N ∞→. For a class of general deterministic positive-definite M × M matrices Σ , under some additional assumptions on the distribution of xij 's, we show that the limiting behavior of the largest eigenvalue of W N is universal, via pursuing a Green function comparison strategy raised in [Probab. Theory Related Fields 154 (2012) 341-407, Adv. Math. 229 (2012) 1435-1515] by Erd″os, Yau and Yin for Wigner matrices and extended by Pillai and Yin [Ann. Appl. Probab. 24 (2014) 935-1001] to sample covariance matrices in the null case (&Epsi = I ). Consequently, in the standard complex case (Ex2 ij = 0), combing this universality property and the results known for Gaussian matrices obtained by El Karoui in [Ann. Probab. 35 (2007) 663-714] (nonsingular case) and Onatski in [Ann. Appl. Probab. 18 (2008) 470-490] (singular case), we show that after an appropriate normalization the largest eigenvalue of W N converges weakly to the type 2 Tracy-Widom distribution TW2 . Moreover, in the real case, we show that whenΣ is spiked with a fixed number of subcritical spikes, the type 1 Tracy-Widom limit TW1 holds for the normalized largest eigenvalue of W N , which extends a result of Féral and Péché in [J. Math. Phys. 50 (2009) 073302] to the scenario of nondiagonal Σ and more generally distributed X . In summary, we establish the Tracy-Widom type universality for the largest eigenvalue of generally distributed sample covariance matrices under quite light assumptions on &Sigma . Applications of these limiting results to statistical signal detection and structure recognition of separable covariance matrices are also discussed."}],"publist_id":"5672","issue":"1","day":"01","month":"02","publisher":"Institute of Mathematical Statistics","year":"2015","type":"journal_article","isi":1,"department":[{"_id":"LaEr"}],"publication":"Annals of Statistics","article_processing_charge":"No","status":"public","external_id":{"arxiv":["1304.5690"],"isi":["000349738500014"]},"date_updated":"2025-09-29T11:02:34Z","language":[{"iso":"eng"}]},{"publication_status":"published","oa_version":"Preprint","title":"The logarithmic law of random determinant","abstract":[{"lang":"eng","text":"Consider the square random matrix An = (aij)n,n, where {aij:= a(n)ij , i, j = 1, . . . , n} is a collection of independent real random variables with means zero and variances one. Under the additional moment condition supn max1≤i,j ≤n Ea4ij <∞, we prove Girko's logarithmic law of det An in the sense that as n→∞ log | detAn| ? (1/2) log(n-1)! d/→√(1/2) log n N(0, 1)."}],"quality_controlled":"1","page":"1600 - 1628","main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1208.5823"}],"volume":21,"intvolume":" 21","oa":1,"date_published":"2015-08-01T00:00:00Z","arxiv":1,"citation":{"ista":"Bao Z, Pan G, Zhou W. 2015. The logarithmic law of random determinant. Bernoulli. 21(3), 1600–1628.","ieee":"Z. Bao, G. Pan, and W. Zhou, “The logarithmic law of random determinant,” Bernoulli, vol. 21, no. 3. Bernoulli Society for Mathematical Statistics and Probability, pp. 1600–1628, 2015.","ama":"Bao Z, Pan G, Zhou W. The logarithmic law of random determinant. Bernoulli. 2015;21(3):1600-1628. doi:10.3150/14-BEJ615","short":"Z. Bao, G. Pan, W. Zhou, Bernoulli 21 (2015) 1600–1628.","mla":"Bao, Zhigang, et al. “The Logarithmic Law of Random Determinant.” Bernoulli, vol. 21, no. 3, Bernoulli Society for Mathematical Statistics and Probability, 2015, pp. 1600–28, doi:10.3150/14-BEJ615.","chicago":"Bao, Zhigang, Guangming Pan, and Wang Zhou. “The Logarithmic Law of Random Determinant.” Bernoulli. Bernoulli Society for Mathematical Statistics and Probability, 2015. https://doi.org/10.3150/14-BEJ615.","apa":"Bao, Z., Pan, G., & Zhou, W. (2015). The logarithmic law of random determinant. Bernoulli. Bernoulli Society for Mathematical Statistics and Probability. https://doi.org/10.3150/14-BEJ615"},"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","author":[{"last_name":"Bao","orcid":"0000-0003-3036-1475","first_name":"Zhigang","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","full_name":"Bao, Zhigang"},{"full_name":"Pan, Guangming","last_name":"Pan","first_name":"Guangming"},{"full_name":"Zhou, Wang","last_name":"Zhou","first_name":"Wang"}],"doi":"10.3150/14-BEJ615","date_created":"2018-12-11T11:52:25Z","_id":"1506","date_updated":"2025-09-23T13:59:56Z","language":[{"iso":"eng"}],"article_processing_charge":"No","publication":"Bernoulli","department":[{"_id":"LaEr"}],"isi":1,"external_id":{"isi":["000356993100012"],"arxiv":["1208.5823"]},"status":"public","type":"journal_article","year":"2015","issue":"3","publist_id":"5671","day":"01","publisher":"Bernoulli Society for Mathematical Statistics and Probability","month":"08"},{"date_updated":"2025-09-23T09:08:38Z","language":[{"iso":"eng"}],"article_processing_charge":"No","publication":"Journal of the European Mathematical Society","isi":1,"department":[{"_id":"LaEr"}],"external_id":{"isi":["000360822900003"],"arxiv":["1211.3786"]},"status":"public","type":"journal_article","year":"2015","day":"01","publist_id":"5669","issue":"8","publisher":"European Mathematical Society","month":"08","publication_status":"published","oa_version":"Preprint","title":"Gap universality of generalized Wigner and β ensembles","abstract":[{"lang":"eng","text":"We consider generalized Wigner ensembles and general β-ensembles with analytic potentials for any β ≥ 1. The recent universality results in particular assert that the local averages of consecutive eigenvalue gaps in the bulk of the spectrum are universal in the sense that they coincide with those of the corresponding Gaussian β-ensembles. In this article, we show that local averaging is not necessary for this result, i.e. we prove that the single gap distributions in the bulk are universal. In fact, with an additional step, our result can be extended to any C4(ℝ) potential."}],"quality_controlled":"1","page":"1927 - 2036","main_file_link":[{"url":"http://arxiv.org/abs/1211.3786","open_access":"1"}],"volume":17,"intvolume":" 17","date_published":"2015-08-01T00:00:00Z","oa":1,"citation":{"mla":"Erdös, László, and Horng Yau. “Gap Universality of Generalized Wigner and β Ensembles.” Journal of the European Mathematical Society, vol. 17, no. 8, European Mathematical Society, 2015, pp. 1927–2036, doi:10.4171/JEMS/548.","chicago":"Erdös, László, and Horng Yau. “Gap Universality of Generalized Wigner and β Ensembles.” Journal of the European Mathematical Society. European Mathematical Society, 2015. https://doi.org/10.4171/JEMS/548.","short":"L. Erdös, H. Yau, Journal of the European Mathematical Society 17 (2015) 1927–2036.","ama":"Erdös L, Yau H. Gap universality of generalized Wigner and β ensembles. Journal of the European Mathematical Society. 2015;17(8):1927-2036. doi:10.4171/JEMS/548","apa":"Erdös, L., & Yau, H. (2015). Gap universality of generalized Wigner and β ensembles. Journal of the European Mathematical Society. European Mathematical Society. https://doi.org/10.4171/JEMS/548","ista":"Erdös L, Yau H. 2015. Gap universality of generalized Wigner and β ensembles. Journal of the European Mathematical Society. 17(8), 1927–2036.","ieee":"L. Erdös and H. Yau, “Gap universality of generalized Wigner and β ensembles,” Journal of the European Mathematical Society, vol. 17, no. 8. European Mathematical Society, pp. 1927–2036, 2015."},"arxiv":1,"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","author":[{"full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","orcid":"0000-0001-5366-9603","last_name":"Erdös"},{"last_name":"Yau","first_name":"Horng","full_name":"Yau, Horng"}],"date_created":"2018-12-11T11:52:26Z","doi":"10.4171/JEMS/548","_id":"1508","scopus_import":"1"},{"language":[{"iso":"eng"}],"date_updated":"2025-09-23T13:39:37Z","status":"public","external_id":{"isi":["000348303100008"],"arxiv":["1309.5106"]},"department":[{"_id":"LaEr"}],"isi":1,"publication":"Communications in Mathematical Physics","article_processing_charge":"No","year":"2015","type":"journal_article","month":"02","publisher":"Springer","issue":"3","day":"01","publist_id":"4818","abstract":[{"lang":"eng","text":"We consider the spectral statistics of large random band matrices on mesoscopic energy scales. We show that the correlation function of the local eigenvalue density exhibits a universal power law behaviour that differs from the Wigner-Dyson- Mehta statistics. This law had been predicted in the physics literature by Altshuler and Shklovskii in (Zh Eksp Teor Fiz (Sov Phys JETP) 91(64):220(127), 1986); it describes the correlations of the eigenvalue density in general metallic sampleswith weak disorder. Our result rigorously establishes the Altshuler-Shklovskii formulas for band matrices. In two dimensions, where the leading term vanishes owing to an algebraic cancellation, we identify the first non-vanishing term and show that it differs substantially from the prediction of Kravtsov and Lerner in (Phys Rev Lett 74:2563-2566, 1995). The proof is given in the current paper and its companion (Ann. H. Poincaré. arXiv:1309.5107, 2014). "}],"oa_version":"Preprint","title":"The Altshuler-Shklovskii formulas for random band matrices I: the unimodular case","publication_status":"published","main_file_link":[{"url":"http://arxiv.org/abs/1309.5106","open_access":"1"}],"page":"1365 - 1416","quality_controlled":"1","intvolume":" 333","volume":333,"scopus_import":"1","_id":"2166","doi":"10.1007/s00220-014-2119-5","date_created":"2018-12-11T11:56:05Z","author":[{"first_name":"László","orcid":"0000-0001-5366-9603","last_name":"Erdös","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Antti","last_name":"Knowles","full_name":"Knowles, Antti"}],"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","arxiv":1,"citation":{"apa":"Erdös, L., & Knowles, A. (2015). The Altshuler-Shklovskii formulas for random band matrices I: the unimodular case. Communications in Mathematical Physics. Springer. https://doi.org/10.1007/s00220-014-2119-5","ama":"Erdös L, Knowles A. The Altshuler-Shklovskii formulas for random band matrices I: the unimodular case. Communications in Mathematical Physics. 2015;333(3):1365-1416. doi:10.1007/s00220-014-2119-5","mla":"Erdös, László, and Antti Knowles. “The Altshuler-Shklovskii Formulas for Random Band Matrices I: The Unimodular Case.” Communications in Mathematical Physics, vol. 333, no. 3, Springer, 2015, pp. 1365–416, doi:10.1007/s00220-014-2119-5.","chicago":"Erdös, László, and Antti Knowles. “The Altshuler-Shklovskii Formulas for Random Band Matrices I: The Unimodular Case.” Communications in Mathematical Physics. Springer, 2015. https://doi.org/10.1007/s00220-014-2119-5.","short":"L. Erdös, A. Knowles, Communications in Mathematical Physics 333 (2015) 1365–1416.","ieee":"L. Erdös and A. Knowles, “The Altshuler-Shklovskii formulas for random band matrices I: the unimodular case,” Communications in Mathematical Physics, vol. 333, no. 3. Springer, pp. 1365–1416, 2015.","ista":"Erdös L, Knowles A. 2015. The Altshuler-Shklovskii formulas for random band matrices I: the unimodular case. Communications in Mathematical Physics. 333(3), 1365–1416."},"date_published":"2015-02-01T00:00:00Z","oa":1},{"publication_status":"published","title":"Asymptotic mutual information statistics of MIMO channels and CLT of sample covariance matrices","oa_version":"None","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)."}],"quality_controlled":"1","page":"3413 - 3426","volume":61,"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","citation":{"apa":"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. IEEE. https://doi.org/10.1109/TIT.2015.2421894","chicago":"Bao, Zhigang, Guangming Pan, and Wang Zhou. “Asymptotic Mutual Information Statistics of MIMO Channels and CLT of Sample Covariance Matrices.” IEEE Transactions on Information Theory. IEEE, 2015. https://doi.org/10.1109/TIT.2015.2421894.","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.” IEEE Transactions on Information Theory, vol. 61, no. 6, IEEE, 2015, pp. 3413–26, doi:10.1109/TIT.2015.2421894.","ama":"Bao Z, Pan G, Zhou W. Asymptotic mutual information statistics of MIMO channels and CLT of sample covariance matrices. IEEE Transactions on Information Theory. 2015;61(6):3413-3426. doi:10.1109/TIT.2015.2421894","ieee":"Z. Bao, G. Pan, and W. Zhou, “Asymptotic mutual information statistics of MIMO channels and CLT of sample covariance matrices,” IEEE Transactions on Information Theory, 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."},"date_published":"2015-06-01T00:00:00Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","author":[{"first_name":"Zhigang","last_name":"Bao","orcid":"0000-0003-3036-1475","full_name":"Bao, Zhigang","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Pan, Guangming","last_name":"Pan","first_name":"Guangming"},{"full_name":"Zhou, Wang","first_name":"Wang","last_name":"Zhou"}],"date_created":"2018-12-11T11:52:52Z","doi":"10.1109/TIT.2015.2421894","scopus_import":"1","_id":"1585","date_updated":"2025-09-23T07:57:31Z","language":[{"iso":"eng"}],"article_processing_charge":"No","department":[{"_id":"LaEr"}],"isi":1,"publication":"IEEE Transactions on Information Theory","external_id":{"isi":["000354943600029"]},"status":"public","type":"journal_article","year":"2015","day":"01","issue":"6","publist_id":"5586","publisher":"IEEE","month":"06"},{"doi":"10.1142/S0129055X1550018X","date_created":"2018-12-11T11:53:24Z","author":[{"full_name":"Lee, Jioon","first_name":"Jioon","last_name":"Lee"},{"full_name":"Schnelli, Kevin","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","first_name":"Kevin","orcid":"0000-0003-0954-3231","last_name":"Schnelli"}],"_id":"1674","scopus_import":"1","date_published":"2015-09-01T00:00:00Z","oa":1,"citation":{"ieee":"J. Lee and K. Schnelli, “Edge universality for deformed Wigner matrices,” Reviews in Mathematical Physics, 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.","apa":"Lee, J., & Schnelli, K. (2015). Edge universality for deformed Wigner matrices. Reviews in Mathematical Physics. World Scientific Publishing. https://doi.org/10.1142/S0129055X1550018X","ama":"Lee J, Schnelli K. Edge universality for deformed Wigner matrices. Reviews in Mathematical Physics. 2015;27(8). doi:10.1142/S0129055X1550018X","mla":"Lee, Jioon, and Kevin Schnelli. “Edge Universality for Deformed Wigner Matrices.” Reviews in Mathematical Physics, vol. 27, no. 8, 1550018, World Scientific Publishing, 2015, doi:10.1142/S0129055X1550018X.","short":"J. Lee, K. Schnelli, Reviews in Mathematical Physics 27 (2015).","chicago":"Lee, Jioon, and Kevin Schnelli. “Edge Universality for Deformed Wigner Matrices.” Reviews in Mathematical Physics. World Scientific Publishing, 2015. https://doi.org/10.1142/S0129055X1550018X."},"arxiv":1,"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","intvolume":" 27","volume":27,"main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1407.8015"}],"quality_controlled":"1","abstract":[{"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.","lang":"eng"}],"publication_status":"published","oa_version":"Preprint","title":"Edge universality for deformed Wigner matrices","publisher":"World Scientific Publishing","month":"09","day":"01","publist_id":"5475","issue":"8","type":"journal_article","year":"2015","external_id":{"arxiv":["1407.8015"],"isi":["000362566600001"]},"article_number":"1550018","status":"public","article_processing_charge":"No","publication":"Reviews in Mathematical Physics","isi":1,"department":[{"_id":"LaEr"}],"language":[{"iso":"eng"}],"date_updated":"2025-09-23T08:32:19Z"},{"oa_version":"Preprint","title":"The local semicircle law for random matrices with a fourfold symmetry","publication_status":"published","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"}],"quality_controlled":"1","main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1506.04683"}],"volume":56,"intvolume":" 56","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","arxiv":1,"citation":{"chicago":"Alt, Johannes. “The Local Semicircle Law for Random Matrices with a Fourfold Symmetry.” Journal of Mathematical Physics. American Institute of Physics, 2015. https://doi.org/10.1063/1.4932606.","mla":"Alt, Johannes. “The Local Semicircle Law for Random Matrices with a Fourfold Symmetry.” Journal of Mathematical Physics, vol. 56, no. 10, 103301, American Institute of Physics, 2015, doi:10.1063/1.4932606.","short":"J. Alt, Journal of Mathematical Physics 56 (2015).","ama":"Alt J. The local semicircle law for random matrices with a fourfold symmetry. Journal of Mathematical Physics. 2015;56(10). doi:10.1063/1.4932606","apa":"Alt, J. (2015). The local semicircle law for random matrices with a fourfold symmetry. Journal of Mathematical Physics. American Institute of Physics. https://doi.org/10.1063/1.4932606","ista":"Alt J. 2015. The local semicircle law for random matrices with a fourfold symmetry. Journal of Mathematical Physics. 56(10), 103301.","ieee":"J. Alt, “The local semicircle law for random matrices with a fourfold symmetry,” Journal of Mathematical Physics, vol. 56, no. 10. American Institute of Physics, 2015."},"date_published":"2015-10-09T00:00:00Z","oa":1,"scopus_import":"1","_id":"1677","doi":"10.1063/1.4932606","author":[{"last_name":"Alt","first_name":"Johannes","id":"36D3D8B6-F248-11E8-B48F-1D18A9856A87","full_name":"Alt, Johannes"}],"date_created":"2018-12-11T11:53:25Z","date_updated":"2026-04-08T14:11:36Z","language":[{"iso":"eng"}],"related_material":{"record":[{"id":"149","status":"public","relation":"dissertation_contains"}]},"department":[{"_id":"LaEr"}],"isi":1,"publication":"Journal of Mathematical Physics","article_processing_charge":"No","status":"public","article_number":"103301","external_id":{"isi":["000364237000026"],"arxiv":["1506.04683"]},"year":"2015","project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804","name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7"}],"type":"journal_article","ec_funded":1,"day":"09","publist_id":"5472","issue":"10","month":"10","corr_author":"1","publisher":"American Institute of Physics"},{"abstract":[{"text":"We prove the universality of the β-ensembles with convex analytic potentials and for any β >\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"}],"publication_status":"published","oa_version":"Preprint","title":"Universality of general β-ensembles","page":"1127 - 1190","main_file_link":[{"url":"http://arxiv.org/abs/1104.2272","open_access":"1"}],"quality_controlled":"1","intvolume":" 163","volume":163,"date_created":"2018-12-11T11:59:08Z","author":[{"orcid":"0000-0001-5366-9603","last_name":"Erdös","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László"},{"first_name":"Paul","last_name":"Bourgade","full_name":"Bourgade, Paul"},{"last_name":"Yau","first_name":"Horng","full_name":"Yau, Horng"}],"doi":"10.1215/00127094-2649752","_id":"2699","scopus_import":"1","date_published":"2014-04-01T00:00:00Z","oa":1,"arxiv":1,"citation":{"apa":"Erdös, L., Bourgade, P., & Yau, H. (2014). Universality of general β-ensembles. Duke Mathematical Journal. Duke University Press. https://doi.org/10.1215/00127094-2649752","chicago":"Erdös, László, Paul Bourgade, and Horng Yau. “Universality of General β-Ensembles.” Duke Mathematical Journal. Duke University Press, 2014. https://doi.org/10.1215/00127094-2649752.","mla":"Erdös, László, et al. “Universality of General β-Ensembles.” Duke Mathematical Journal, vol. 163, no. 6, Duke University Press, 2014, pp. 1127–90, doi:10.1215/00127094-2649752.","short":"L. Erdös, P. Bourgade, H. Yau, Duke Mathematical Journal 163 (2014) 1127–1190.","ama":"Erdös L, Bourgade P, Yau H. Universality of general β-ensembles. Duke Mathematical Journal. 2014;163(6):1127-1190. doi:10.1215/00127094-2649752","ieee":"L. Erdös, P. Bourgade, and H. Yau, “Universality of general β-ensembles,” Duke Mathematical Journal, 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."},"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","language":[{"iso":"eng"}],"date_updated":"2025-09-29T11:11:17Z","external_id":{"isi":["000334478100003"],"arxiv":["1104.2272"]},"status":"public","article_processing_charge":"No","publication":"Duke Mathematical Journal","department":[{"_id":"LaEr"}],"isi":1,"type":"journal_article","year":"2014","publisher":"Duke University Press","month":"04","day":"01","publist_id":"4197","issue":"6"},{"quality_controlled":"1","page":"441 - 464","main_file_link":[{"url":"http://arxiv.org/abs/1407.1552","open_access":"1"}],"publication_status":"published","title":"Phase transition in the density of states of quantum spin glasses","oa_version":"Submitted Version","abstract":[{"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.","lang":"eng"}],"arxiv":1,"citation":{"apa":"Erdös, L., & Schröder, D. J. (2014). Phase transition in the density of states of quantum spin glasses. Mathematical Physics, Analysis and Geometry. Springer. https://doi.org/10.1007/s11040-014-9164-3","mla":"Erdös, László, and Dominik J. Schröder. “Phase Transition in the Density of States of Quantum Spin Glasses.” Mathematical Physics, Analysis and Geometry, vol. 17, no. 3–4, Springer, 2014, pp. 441–64, doi:10.1007/s11040-014-9164-3.","short":"L. Erdös, D.J. Schröder, Mathematical Physics, Analysis and Geometry 17 (2014) 441–464.","chicago":"Erdös, László, and Dominik J Schröder. “Phase Transition in the Density of States of Quantum Spin Glasses.” Mathematical Physics, Analysis and Geometry. Springer, 2014. https://doi.org/10.1007/s11040-014-9164-3.","ama":"Erdös L, Schröder DJ. Phase transition in the density of states of quantum spin glasses. Mathematical Physics, Analysis and Geometry. 2014;17(3-4):441-464. doi:10.1007/s11040-014-9164-3","ieee":"L. Erdös and D. J. Schröder, “Phase transition in the density of states of quantum spin glasses,” Mathematical Physics, Analysis and Geometry, vol. 17, no. 3–4. Springer, pp. 441–464, 2014.","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."},"date_published":"2014-12-17T00:00:00Z","oa":1,"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_created":"2018-12-11T11:55:15Z","doi":"10.1007/s11040-014-9164-3","author":[{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László","orcid":"0000-0001-5366-9603","last_name":"Erdös","first_name":"László"},{"first_name":"Dominik J","last_name":"Schröder","full_name":"Schröder, Dominik J"}],"scopus_import":"1","_id":"2019","volume":17,"intvolume":" 17","article_processing_charge":"No","isi":1,"department":[{"_id":"LaEr"}],"publication":"Mathematical Physics, Analysis and Geometry","external_id":{"arxiv":["1407.1552"],"isi":["000348286700011"]},"status":"public","date_updated":"2025-09-29T12:00:05Z","language":[{"iso":"eng"}],"issue":"3-4","publist_id":"5053","day":"17","publisher":"Springer","month":"12","project":[{"name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7","grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425"}],"type":"journal_article","year":"2014","ec_funded":1},{"intvolume":" 17","article_type":"original","volume":17,"date_created":"2018-12-11T11:54:45Z","author":[{"orcid":"0000-0001-8255-3968","last_name":"Sadel","first_name":"Christian","id":"4760E9F8-F248-11E8-B48F-1D18A9856A87","full_name":"Sadel, Christian"}],"doi":"10.1007/s11040-014-9163-4","scopus_import":"1","_id":"1926","citation":{"chicago":"Sadel, Christian. “Absolutely Continuous Spectrum for Random Schrödinger Operators on the Fibonacci and Similar Tree-Strips.” Mathematical Physics, Analysis and Geometry. Springer, 2014. https://doi.org/10.1007/s11040-014-9163-4.","mla":"Sadel, Christian. “Absolutely Continuous Spectrum for Random Schrödinger Operators on the Fibonacci and Similar Tree-Strips.” Mathematical Physics, Analysis and Geometry, vol. 17, no. 3–4, Springer, 2014, pp. 409–40, doi:10.1007/s11040-014-9163-4.","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. Mathematical Physics, Analysis and Geometry. 2014;17(3-4):409-440. doi:10.1007/s11040-014-9163-4","apa":"Sadel, C. (2014). Absolutely continuous spectrum for random Schrödinger operators on the Fibonacci and similar Tree-strips. Mathematical Physics, Analysis and Geometry. Springer. https://doi.org/10.1007/s11040-014-9163-4","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.","ieee":"C. Sadel, “Absolutely continuous spectrum for random Schrödinger operators on the Fibonacci and similar Tree-strips,” Mathematical Physics, Analysis and Geometry, vol. 17, no. 3–4. Springer, pp. 409–440, 2014."},"arxiv":1,"oa":1,"date_published":"2014-12-17T00:00:00Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","abstract":[{"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.","lang":"eng"}],"publication_status":"published","title":"Absolutely continuous spectrum for random Schrödinger operators on the Fibonacci and similar Tree-strips","oa_version":"Preprint","page":"409 - 440","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1304.3862"}],"quality_controlled":"1","ec_funded":1,"project":[{"_id":"26450934-B435-11E9-9278-68D0E5697425","name":"NSERC Postdoctoral fellowship"},{"_id":"25681D80-B435-11E9-9278-68D0E5697425","grant_number":"291734","name":"International IST Postdoc Fellowship Programme","call_identifier":"FP7"}],"type":"journal_article","year":"2014","publisher":"Springer","month":"12","corr_author":"1","issue":"3-4","publist_id":"5168","day":"17","language":[{"iso":"eng"}],"date_updated":"2025-09-29T12:14:12Z","external_id":{"isi":["000348286700010"],"arxiv":["1304.3862"]},"status":"public","article_processing_charge":"No","isi":1,"department":[{"_id":"LaEr"}],"publication":"Mathematical Physics, Analysis and Geometry"},{"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_published":"2014-11-01T00:00:00Z","oa":1,"arxiv":1,"citation":{"ama":"Bourgade P, Erdös L, Yau H. Edge universality of beta ensembles. Communications in Mathematical Physics. 2014;332(1):261-353. doi:10.1007/s00220-014-2120-z","chicago":"Bourgade, Paul, László Erdös, and Horngtzer Yau. “Edge Universality of Beta Ensembles.” Communications in Mathematical Physics. Springer, 2014. https://doi.org/10.1007/s00220-014-2120-z.","mla":"Bourgade, Paul, et al. “Edge Universality of Beta Ensembles.” Communications in Mathematical Physics, vol. 332, no. 1, Springer, 2014, pp. 261–353, doi:10.1007/s00220-014-2120-z.","short":"P. Bourgade, L. Erdös, H. Yau, Communications in Mathematical Physics 332 (2014) 261–353.","apa":"Bourgade, P., Erdös, L., & Yau, H. (2014). Edge universality of beta ensembles. Communications in Mathematical Physics. Springer. https://doi.org/10.1007/s00220-014-2120-z","ista":"Bourgade P, Erdös L, Yau H. 2014. Edge universality of beta ensembles. Communications in Mathematical Physics. 332(1), 261–353.","ieee":"P. Bourgade, L. Erdös, and H. Yau, “Edge universality of beta ensembles,” Communications in Mathematical Physics, vol. 332, no. 1. Springer, pp. 261–353, 2014."},"_id":"1937","scopus_import":"1","author":[{"full_name":"Bourgade, Paul","last_name":"Bourgade","first_name":"Paul"},{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László","last_name":"Erdös","orcid":"0000-0001-5366-9603","first_name":"László"},{"full_name":"Yau, Horngtzer","first_name":"Horngtzer","last_name":"Yau"}],"doi":"10.1007/s00220-014-2120-z","date_created":"2018-12-11T11:54:48Z","volume":332,"intvolume":" 332","quality_controlled":"1","main_file_link":[{"url":"http://arxiv.org/abs/1306.5728","open_access":"1"}],"page":"261 - 353","oa_version":"Submitted Version","title":"Edge universality of beta ensembles","publication_status":"published","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"}],"day":"01","publist_id":"5158","issue":"1","month":"11","publisher":"Springer","year":"2014","project":[{"_id":"25BDE9A4-B435-11E9-9278-68D0E5697425","name":"Glutamaterge synaptische Ãbertragung und Plastizität in hippocampalen Mikroschaltkreisen","grant_number":"SFB-TR3-TP10B"}],"type":"journal_article","publication":"Communications in Mathematical Physics","department":[{"_id":"LaEr"}],"isi":1,"article_processing_charge":"No","status":"public","external_id":{"isi":["000341491700007"],"arxiv":["1306.5728"]},"date_updated":"2025-09-29T12:08:21Z","language":[{"iso":"eng"}]},{"page":"214 - 236","main_file_link":[{"url":"http://arxiv.org/abs/1407.5752","open_access":"1"}],"quality_controlled":"1","abstract":[{"lang":"eng","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."}],"publication_status":"published","title":"Random matrices, log-gases and Hölder regularity","oa_version":"Submitted Version","author":[{"full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","first_name":"László","orcid":"0000-0001-5366-9603","last_name":"Erdös"}],"date_created":"2018-12-11T11:52:25Z","_id":"1507","scopus_import":"1","oa":1,"date_published":"2014-08-01T00:00:00Z","arxiv":1,"citation":{"ama":"Erdös L. Random matrices, log-gases and Hölder regularity. In: Proceedings of the International Congress of Mathematicians. Vol 3. International Congress of Mathematicians; 2014:214-236.","mla":"Erdös, László. “Random Matrices, Log-Gases and Hölder Regularity.” Proceedings of the International Congress of Mathematicians, vol. 3, International Congress of Mathematicians, 2014, pp. 214–36.","short":"L. Erdös, in:, Proceedings of the International Congress of Mathematicians, International Congress of Mathematicians, 2014, pp. 214–236.","chicago":"Erdös, László. “Random Matrices, Log-Gases and Hölder Regularity.” In Proceedings of the International Congress of Mathematicians, 3:214–36. International Congress of Mathematicians, 2014.","apa":"Erdös, L. (2014). Random matrices, log-gases and Hölder regularity. In Proceedings of the International Congress of Mathematicians (Vol. 3, pp. 214–236). Seoul, Korea: International Congress of Mathematicians.","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.","ieee":"L. Erdös, “Random matrices, log-gases and Hölder regularity,” in Proceedings of the International Congress of Mathematicians, Seoul, Korea, 2014, vol. 3, pp. 214–236."},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","intvolume":" 3","acknowledgement":"The author is partially supported by SFB-TR 12 Grant of the German Research Council.","volume":3,"external_id":{"arxiv":["1407.5752"]},"status":"public","article_processing_charge":"No","publication":"Proceedings of the International Congress of Mathematicians","department":[{"_id":"LaEr"}],"language":[{"iso":"eng"}],"date_updated":"2025-06-11T07:52:59Z","conference":{"start_date":"2014-08-13","name":"ICM: International Congress of Mathematicians","end_date":"2014-08-21","location":"Seoul, Korea"},"publisher":"International Congress of Mathematicians","month":"08","publist_id":"5670","day":"01","ec_funded":1,"project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804","name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7"}],"type":"conference","year":"2014"},{"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"year":"2014","type":"journal_article","pubrep_id":"426","corr_author":"1","month":"06","publisher":"Institute of Mathematical Statistics","publist_id":"4803","day":"09","language":[{"iso":"eng"}],"date_updated":"2025-09-29T11:35:54Z","ddc":["570"],"file":[{"relation":"main_file","creator":"system","access_level":"open_access","checksum":"bd8a041c76d62fe820bf73ff13ce7d1b","file_id":"4729","file_name":"IST-2016-426-v1+1_3121-17518-1-PB.pdf","date_created":"2018-12-12T10:09:06Z","content_type":"application/pdf","date_updated":"2020-07-14T12:45:31Z","file_size":327322}],"status":"public","external_id":{"isi":["000341869300001"]},"publication":"Electronic Communications in Probability","department":[{"_id":"LaEr"}],"has_accepted_license":"1","isi":1,"article_processing_charge":"No","intvolume":" 19","volume":19,"_id":"2179","scopus_import":"1","doi":"10.1214/ECP.v19-3121","author":[{"last_name":"Ajanki","first_name":"Oskari H","id":"36F2FB7E-F248-11E8-B48F-1D18A9856A87","full_name":"Ajanki, Oskari H"},{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László","last_name":"Erdös","orcid":"0000-0001-5366-9603","first_name":"László"},{"id":"3020C786-F248-11E8-B48F-1D18A9856A87","full_name":"Krüger, Torben H","last_name":"Krüger","orcid":"0000-0002-4821-3297","first_name":"Torben H"}],"date_created":"2018-12-11T11:56:10Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","oa":1,"date_published":"2014-06-09T00:00:00Z","citation":{"apa":"Ajanki, O. H., Erdös, L., & Krüger, T. H. (2014). Local semicircle law with imprimitive variance matrix. Electronic Communications in Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/ECP.v19-3121","short":"O.H. Ajanki, L. Erdös, T.H. Krüger, Electronic Communications in Probability 19 (2014).","chicago":"Ajanki, Oskari H, László Erdös, and Torben H Krüger. “Local Semicircle Law with Imprimitive Variance Matrix.” Electronic Communications in Probability. Institute of Mathematical Statistics, 2014. https://doi.org/10.1214/ECP.v19-3121.","mla":"Ajanki, Oskari H., et al. “Local Semicircle Law with Imprimitive Variance Matrix.” Electronic Communications in Probability, vol. 19, Institute of Mathematical Statistics, 2014, doi:10.1214/ECP.v19-3121.","ama":"Ajanki OH, Erdös L, Krüger TH. Local semicircle law with imprimitive variance matrix. Electronic Communications in Probability. 2014;19. doi:10.1214/ECP.v19-3121","ieee":"O. H. Ajanki, L. Erdös, and T. H. Krüger, “Local semicircle law with imprimitive variance matrix,” Electronic Communications in Probability, 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."},"abstract":[{"lang":"eng","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."}],"oa_version":"Published Version","title":"Local semicircle law with imprimitive variance matrix","publication_status":"published","quality_controlled":"1","file_date_updated":"2020-07-14T12:45:31Z"},{"day":"15","publist_id":"4739","month":"03","publisher":"Institute of Mathematical Statistics","year":"2014","type":"journal_article","project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems","grant_number":"338804"}],"pubrep_id":"427","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"ec_funded":1,"publication":"Electronic Journal of Probability","isi":1,"has_accepted_license":"1","department":[{"_id":"LaEr"}],"article_processing_charge":"No","status":"public","external_id":{"isi":["000340998500001"]},"article_number":"33","file":[{"content_type":"application/pdf","date_updated":"2020-07-14T12:45:34Z","file_size":810150,"relation":"main_file","access_level":"open_access","creator":"system","file_name":"IST-2016-427-v1+1_3054-16624-4-PB.pdf","date_created":"2018-12-12T10:14:06Z","file_id":"5055","checksum":"7eb297ff367a2ee73b21b6dd1e1948e4"}],"ddc":["510"],"date_updated":"2025-09-29T11:26:42Z","language":[{"iso":"eng"}],"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","oa":1,"date_published":"2014-03-15T00:00:00Z","citation":{"ieee":"A. Bloemendal, L. Erdös, A. Knowles, H. Yau, and J. Yin, “Isotropic local laws for sample covariance and generalized Wigner matrices,” Electronic Journal of Probability, vol. 19. Institute of Mathematical Statistics, 2014.","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.","apa":"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. Institute of Mathematical Statistics. https://doi.org/10.1214/EJP.v19-3054","chicago":"Bloemendal, Alex, László Erdös, Antti Knowles, Horng Yau, and Jun Yin. “Isotropic Local Laws for Sample Covariance and Generalized Wigner Matrices.” Electronic Journal of Probability. Institute of Mathematical Statistics, 2014. https://doi.org/10.1214/EJP.v19-3054.","mla":"Bloemendal, Alex, et al. “Isotropic Local Laws for Sample Covariance and Generalized Wigner Matrices.” Electronic Journal of Probability, vol. 19, 33, Institute of Mathematical Statistics, 2014, doi:10.1214/EJP.v19-3054.","short":"A. Bloemendal, L. Erdös, A. Knowles, H. Yau, J. Yin, Electronic Journal of Probability 19 (2014).","ama":"Bloemendal A, Erdös L, Knowles A, Yau H, Yin J. Isotropic local laws for sample covariance and generalized Wigner matrices. Electronic Journal of Probability. 2014;19. doi:10.1214/EJP.v19-3054"},"_id":"2225","scopus_import":"1","author":[{"full_name":"Bloemendal, Alex","last_name":"Bloemendal","first_name":"Alex"},{"first_name":"László","orcid":"0000-0001-5366-9603","last_name":"Erdös","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Knowles, Antti","first_name":"Antti","last_name":"Knowles"},{"first_name":"Horng","last_name":"Yau","full_name":"Yau, Horng"},{"last_name":"Yin","first_name":"Jun","full_name":"Yin, Jun"}],"doi":"10.1214/EJP.v19-3054","date_created":"2018-12-11T11:56:25Z","volume":19,"publication_identifier":{"issn":["1083-6489"]},"intvolume":" 19","quality_controlled":"1","file_date_updated":"2020-07-14T12:45:34Z","oa_version":"Published Version","title":"Isotropic local laws for sample covariance and generalized Wigner matrices","publication_status":"published","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"}]},{"type":"journal_article","year":"2013","day":"16","issue":"6","publist_id":"4198","publisher":"European Mathematical Society","month":"10","date_updated":"2025-09-29T14:09:03Z","language":[{"iso":"eng"}],"article_processing_charge":"No","publication":"Journal of the European Mathematical Society","department":[{"_id":"LaEr"}],"isi":1,"external_id":{"isi":["000326323400006"],"arxiv":["1105.0506"]},"status":"public","volume":15,"intvolume":" 15","date_published":"2013-10-16T00:00:00Z","oa":1,"citation":{"apa":"Erdös, L., Fournais, S., & Solovej, J. (2013). Stability and semiclassics in self-generated fields. Journal of the European Mathematical Society. European Mathematical Society. https://doi.org/10.4171/JEMS/416","ama":"Erdös L, Fournais S, Solovej J. Stability and semiclassics in self-generated fields. Journal of the European Mathematical Society. 2013;15(6):2093-2113. doi:10.4171/JEMS/416","mla":"Erdös, László, et al. “Stability and Semiclassics in Self-Generated Fields.” Journal of the European Mathematical Society, vol. 15, no. 6, European Mathematical Society, 2013, pp. 2093–113, doi:10.4171/JEMS/416.","chicago":"Erdös, László, Søren Fournais, and Jan Solovej. “Stability and Semiclassics in Self-Generated Fields.” Journal of the European Mathematical Society. European Mathematical Society, 2013. https://doi.org/10.4171/JEMS/416.","short":"L. Erdös, S. Fournais, J. Solovej, Journal of the European Mathematical Society 15 (2013) 2093–2113.","ieee":"L. Erdös, S. Fournais, and J. Solovej, “Stability and semiclassics in self-generated fields,” Journal of the European Mathematical Society, vol. 15, no. 6. European Mathematical Society, 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."},"arxiv":1,"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","doi":"10.4171/JEMS/416","author":[{"first_name":"László","last_name":"Erdös","orcid":"0000-0001-5366-9603","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Søren","last_name":"Fournais","full_name":"Fournais, Søren"},{"full_name":"Solovej, Jan","first_name":"Jan","last_name":"Solovej"}],"date_created":"2018-12-11T11:59:07Z","_id":"2698","publication_status":"published","oa_version":"Preprint","title":"Stability and semiclassics in self-generated fields","abstract":[{"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.","lang":"eng"}],"quality_controlled":"1","page":"2093 - 2113","main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1105.0506"}]},{"status":"public","external_id":{"isi":["000323203800001"],"arxiv":["1207.0031"]},"publication":"Journal of Statistical Physics","isi":1,"department":[{"_id":"LaEr"}],"article_processing_charge":"No","language":[{"iso":"eng"}],"date_updated":"2025-09-29T14:07:43Z","corr_author":"1","month":"07","publisher":"Springer","day":"18","publist_id":"4107","issue":"6","year":"2013","type":"journal_article","main_file_link":[{"url":"http://arxiv.org/abs/1207.0031","open_access":"1"}],"page":"1003 - 1032","quality_controlled":"1","abstract":[{"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.","lang":"eng"}],"oa_version":"Preprint","title":"Local eigenvalue density for general MANOVA matrices","publication_status":"published","_id":"2782","scopus_import":"1","author":[{"last_name":"Erdös","orcid":"0000-0001-5366-9603","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László"},{"last_name":"Farrell","first_name":"Brendan","full_name":"Farrell, Brendan"}],"doi":"10.1007/s10955-013-0807-8","date_created":"2018-12-11T11:59:34Z","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","date_published":"2013-07-18T00:00:00Z","oa":1,"arxiv":1,"citation":{"ama":"Erdös L, Farrell B. Local eigenvalue density for general MANOVA matrices. Journal of Statistical Physics. 2013;152(6):1003-1032. doi:10.1007/s10955-013-0807-8","short":"L. Erdös, B. Farrell, Journal of Statistical Physics 152 (2013) 1003–1032.","chicago":"Erdös, László, and Brendan Farrell. “Local Eigenvalue Density for General MANOVA Matrices.” Journal of Statistical Physics. Springer, 2013. https://doi.org/10.1007/s10955-013-0807-8.","mla":"Erdös, László, and Brendan Farrell. “Local Eigenvalue Density for General MANOVA Matrices.” Journal of Statistical Physics, vol. 152, no. 6, Springer, 2013, pp. 1003–32, doi:10.1007/s10955-013-0807-8.","apa":"Erdös, L., & Farrell, B. (2013). Local eigenvalue density for general MANOVA matrices. Journal of Statistical Physics. Springer. https://doi.org/10.1007/s10955-013-0807-8","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,” Journal of Statistical Physics, vol. 152, no. 6. Springer, pp. 1003–1032, 2013."},"intvolume":" 152","volume":152},{"doi":"10.1214/EJP.v18-2473","date_created":"2018-12-11T11:59:51Z","author":[{"first_name":"László","orcid":"0000-0001-5366-9603","last_name":"Erdös","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Knowles, Antti","first_name":"Antti","last_name":"Knowles"},{"first_name":"Horng","last_name":"Yau","full_name":"Yau, Horng"},{"full_name":"Yin, Jun","first_name":"Jun","last_name":"Yin"}],"_id":"2837","scopus_import":"1","date_published":"2013-05-29T00:00:00Z","oa":1,"citation":{"ama":"Erdös L, Knowles A, Yau H, Yin J. The local semicircle law for a general class of random matrices. Electronic Journal of Probability. 2013;18(59):1-58. doi:10.1214/EJP.v18-2473","mla":"Erdös, László, et al. “The Local Semicircle Law for a General Class of Random Matrices.” Electronic Journal of Probability, vol. 18, no. 59, Institute of Mathematical Statistics, 2013, pp. 1–58, doi:10.1214/EJP.v18-2473.","chicago":"Erdös, László, Antti Knowles, Horng Yau, and Jun Yin. “The Local Semicircle Law for a General Class of Random Matrices.” Electronic Journal of Probability. Institute of Mathematical Statistics, 2013. https://doi.org/10.1214/EJP.v18-2473.","short":"L. Erdös, A. Knowles, H. Yau, J. Yin, Electronic Journal of Probability 18 (2013) 1–58.","apa":"Erdös, L., Knowles, A., Yau, H., & Yin, J. (2013). The local semicircle law for a general class of random matrices. Electronic Journal of Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/EJP.v18-2473","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.","ieee":"L. Erdös, A. Knowles, H. Yau, and J. Yin, “The local semicircle law for a general class of random matrices,” Electronic Journal of Probability, vol. 18, no. 59. Institute of Mathematical Statistics, pp. 1–58, 2013."},"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","intvolume":" 18","volume":18,"page":"1-58","file_date_updated":"2020-07-14T12:45:50Z","quality_controlled":"1","abstract":[{"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 > 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].","lang":"eng"}],"publication_status":"published","oa_version":"Published Version","title":"The local semicircle law for a general class of random matrices","publisher":"Institute of Mathematical Statistics","month":"05","publist_id":"3962","day":"29","issue":"59","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"type":"journal_article","pubrep_id":"406","year":"2013","external_id":{"isi":["000319561600001"]},"status":"public","article_processing_charge":"No","publication":"Electronic Journal of Probability","has_accepted_license":"1","department":[{"_id":"LaEr"}],"isi":1,"language":[{"iso":"eng"}],"file":[{"date_updated":"2020-07-14T12:45:50Z","file_size":651497,"content_type":"application/pdf","relation":"main_file","access_level":"open_access","creator":"system","checksum":"aac9e52a00cb2f5149dc9e362b5ccf44","date_created":"2018-12-12T10:15:46Z","file_name":"IST-2016-406-v1+1_2473-13759-1-PB.pdf","file_id":"5169"}],"ddc":["530"],"date_updated":"2025-09-29T13:46:52Z"}]