[{"issue":"3","project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems","grant_number":"338804","call_identifier":"FP7"}],"month":"03","oa":1,"volume":16,"author":[{"last_name":"Erdös","orcid":"0000-0001-5366-9603","first_name":"László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László"},{"full_name":"Knowles, Antti","last_name":"Knowles","first_name":"Antti"}],"type":"journal_article","ec_funded":1,"scopus_import":1,"publisher":"Springer","date_created":"2018-12-11T11:54:26Z","_id":"1864","citation":{"ieee":"L. Erdös and A. Knowles, “The Altshuler–Shklovskii formulas for random band matrices II: The general case,” *Annales Henri Poincare*, vol. 16, no. 3. Springer, pp. 709–799, 2015.","short":"L. Erdös, A. Knowles, Annales Henri Poincare 16 (2015) 709–799.","ama":"Erdös L, Knowles A. The Altshuler–Shklovskii formulas for random band matrices II: The general case. *Annales Henri Poincare*. 2015;16(3):709-799. doi:10.1007/s00023-014-0333-5","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.","apa":"Erdös, L., & Knowles, A. (2015). The Altshuler–Shklovskii formulas for random band matrices II: The general case. *Annales Henri Poincare*. Springer. https://doi.org/10.1007/s00023-014-0333-5","chicago":"Erdös, László, and Antti Knowles. “The Altshuler–Shklovskii Formulas for Random Band Matrices II: The General Case.” *Annales Henri Poincare*. Springer, 2015. https://doi.org/10.1007/s00023-014-0333-5.","mla":"Erdös, László, and Antti Knowles. “The Altshuler–Shklovskii Formulas for Random Band Matrices II: The General Case.” *Annales Henri Poincare*, vol. 16, no. 3, Springer, 2015, pp. 709–99, doi:10.1007/s00023-014-0333-5."},"year":"2015","intvolume":" 16","publication_status":"published","date_updated":"2021-01-12T06:53:42Z","main_file_link":[{"url":"http://arxiv.org/abs/1309.5107","open_access":"1"}],"title":"The Altshuler–Shklovskii formulas for random band matrices II: The general case","department":[{"_id":"LaEr"}],"page":"709 - 799","date_published":"2015-03-01T00:00:00Z","status":"public","publist_id":"5233","oa_version":"Preprint","doi":"10.1007/s00023-014-0333-5","abstract":[{"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","lang":"eng"}],"publication":"Annales Henri Poincare","day":"01","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}]},{"author":[{"first_name":"László","orcid":"0000-0001-5366-9603","last_name":"Erdös","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László"},{"first_name":"Antti","last_name":"Knowles","full_name":"Knowles, Antti"}],"type":"journal_article","doi":"10.1007/s00220-014-2119-5","abstract":[{"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). ","lang":"eng"}],"oa_version":"Preprint","oa":1,"volume":333,"month":"02","department":[{"_id":"LaEr"}],"title":"The Altshuler-Shklovskii formulas for random band matrices I: the unimodular case","page":"1365 - 1416","date_published":"2015-02-01T00:00:00Z","status":"public","publist_id":"4818","publication_status":"published","issue":"3","date_updated":"2021-01-12T06:55:43Z","quality_controlled":"1","main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1309.5106"}],"day":"01","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}],"intvolume":" 333","year":"2015","publication":"Communications in Mathematical Physics","publisher":"Springer","scopus_import":1,"date_created":"2018-12-11T11:56:05Z","citation":{"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.","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","short":"L. Erdös, A. Knowles, Communications in Mathematical Physics 333 (2015) 1365–1416.","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","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.","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.","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."},"_id":"2166"},{"year":"2015","publication":"Annals of Statistics","_id":"1505","citation":{"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","short":"Z. Bao, G. Pan, W. Zhou, Annals of Statistics 43 (2015) 382–421.","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.","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.","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.","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.","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"},"publisher":"Institute of Mathematical Statistics","date_created":"2018-12-11T11:52:25Z","intvolume":" 43","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","day":"01","language":[{"iso":"eng"}],"status":"public","publist_id":"5672","title":"Universality for the largest eigenvalue of sample covariance matrices with general population","month":"02","department":[{"_id":"LaEr"}],"page":"382 - 421","date_published":"2015-02-01T00:00:00Z","issue":"1","quality_controlled":"1","main_file_link":[{"url":"https://arxiv.org/abs/1304.5690","open_access":"1"}],"date_updated":"2021-01-12T06:51:14Z","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."}],"author":[{"id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","first_name":"Zhigang","orcid":"0000-0003-3036-1475","last_name":"Bao","full_name":"Bao, Zhigang"},{"first_name":"Guangming","last_name":"Pan","full_name":"Pan, Guangming"},{"first_name":"Wang","last_name":"Zhou","full_name":"Zhou, Wang"}],"type":"journal_article","doi":"10.1214/14-AOS1281","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","oa":1,"oa_version":"Preprint","volume":43},{"day":"01","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}],"intvolume":" 21","year":"2015","publication":"Bernoulli","publisher":"Bernoulli Society for Mathematical Statistics and Probability","date_created":"2018-12-11T11:52:25Z","citation":{"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.","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","ista":"Bao Z, Pan G, Zhou W. 2015. The logarithmic law of random determinant. Bernoulli. 21(3), 1600–1628.","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.","short":"Z. Bao, G. Pan, W. Zhou, Bernoulli 21 (2015) 1600–1628.","ama":"Bao Z, Pan G, Zhou W. The logarithmic law of random determinant. *Bernoulli*. 2015;21(3):1600-1628. doi:10.3150/14-BEJ615","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."},"_id":"1506","author":[{"full_name":"Bao, Zhigang","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","last_name":"Bao","orcid":"0000-0003-3036-1475","first_name":"Zhigang"},{"full_name":"Pan, Guangming","first_name":"Guangming","last_name":"Pan"},{"last_name":"Zhou","first_name":"Wang","full_name":"Zhou, Wang"}],"doi":"10.3150/14-BEJ615","type":"journal_article","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 <∞, 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"}],"oa":1,"oa_version":"Preprint","volume":21,"title":"The logarithmic law of random determinant","month":"08","department":[{"_id":"LaEr"}],"date_published":"2015-08-01T00:00:00Z","page":"1600 - 1628","publist_id":"5671","status":"public","publication_status":"published","issue":"3","quality_controlled":"1","main_file_link":[{"url":"http://arxiv.org/abs/1208.5823","open_access":"1"}],"date_updated":"2021-01-12T06:51:14Z"},{"_id":"1508","citation":{"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","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.","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.","ista":"Erdös L, Yau H. 2015. Gap universality of generalized Wigner and β ensembles. Journal of the European Mathematical Society. 17(8), 1927–2036.","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"},"publisher":"European Mathematical Society","scopus_import":1,"date_created":"2018-12-11T11:52:26Z","year":"2015","publication":"Journal of the European Mathematical Society","intvolume":" 17","day":"01","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}],"issue":"8","date_updated":"2021-01-12T06:51:15Z","quality_controlled":"1","main_file_link":[{"url":"http://arxiv.org/abs/1211.3786","open_access":"1"}],"publication_status":"published","publist_id":"5669","status":"public","title":"Gap universality of generalized Wigner and β ensembles","month":"08","department":[{"_id":"LaEr"}],"date_published":"2015-08-01T00:00:00Z","page":"1927 - 2036","oa_version":"Preprint","oa":1,"volume":17,"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."}],"author":[{"full_name":"Erdös, László","first_name":"László","orcid":"0000-0001-5366-9603","last_name":"Erdös","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Yau","first_name":"Horng","full_name":"Yau, Horng"}],"doi":"10.4171/JEMS/548","type":"journal_article"}]