[{"date_updated":"2021-01-12T08:11:24Z","volume":15,"user_id":"D865714E-FA4E-11E9-B85B-F5C5E5697425","department":[{"_id":"LaEr"},{"_id":"JaMa"}],"article_type":"original","status":"public","external_id":{"arxiv":["1705.08836"]},"title":"Transition to shocks in TASEP and decoupling of last passage times","publication":"Latin American Journal of Probability and Mathematical Statistics","file":[{"relation":"main_file","creator":"kschuh","file_id":"5981","access_level":"open_access","file_name":"2018_ALEA_Nejjar.pdf","file_size":394851,"date_updated":"2020-07-14T12:47:46Z","date_created":"2019-02-14T09:44:10Z","content_type":"application/pdf","checksum":"2ded46aa284a836a8cbb34133a64f1cb"}],"_id":"70","project":[{"name":"Random matrices, universality and disordered quantum systems","call_identifier":"FP7","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804"},{"name":"Optimal Transport and Stochastic Dynamics","call_identifier":"H2020","grant_number":"716117","_id":"256E75B8-B435-11E9-9278-68D0E5697425"}],"scopus_import":1,"language":[{"iso":"eng"}],"abstract":[{"lang":"eng","text":"We consider the totally asymmetric simple exclusion process in a critical scaling parametrized by a≥0, which creates a shock in the particle density of order aT−1/3, T the observation time. When starting from step initial data, we provide bounds on the limiting law which in particular imply that in the double limit lima→∞limT→∞ one recovers the product limit law and the degeneration of the correlation length observed at shocks of order 1. This result is shown to apply to a general last-passage percolation model. We also obtain bounds on the two-point functions of several airy processes."}],"ddc":["510"],"page":"1311-1334","issue":"2","oa":1,"date_created":"2018-12-11T11:44:28Z","publication_status":"published","publisher":"ALEA","doi":"10.30757/ALEA.v15-49","oa_version":"Published Version","article_processing_charge":"No","ec_funded":1,"quality_controlled":"1","intvolume":" 15","day":"01","month":"10","has_accepted_license":"1","citation":{"chicago":"Nejjar, Peter. “Transition to Shocks in TASEP and Decoupling of Last Passage Times.” *Latin American Journal of Probability and Mathematical Statistics*. ALEA, 2018. https://doi.org/10.30757/ALEA.v15-49.","short":"P. Nejjar, Latin American Journal of Probability and Mathematical Statistics 15 (2018) 1311–1334.","ama":"Nejjar P. Transition to shocks in TASEP and decoupling of last passage times. *Latin American Journal of Probability and Mathematical Statistics*. 2018;15(2):1311-1334. doi:10.30757/ALEA.v15-49","ieee":"P. Nejjar, “Transition to shocks in TASEP and decoupling of last passage times,” *Latin American Journal of Probability and Mathematical Statistics*, vol. 15, no. 2. ALEA, pp. 1311–1334, 2018.","ista":"Nejjar P. 2018. Transition to shocks in TASEP and decoupling of last passage times. Latin American Journal of Probability and Mathematical Statistics. 15(2), 1311–1334.","apa":"Nejjar, P. (2018). Transition to shocks in TASEP and decoupling of last passage times. *Latin American Journal of Probability and Mathematical Statistics*. ALEA. https://doi.org/10.30757/ALEA.v15-49","mla":"Nejjar, Peter. “Transition to Shocks in TASEP and Decoupling of Last Passage Times.” *Latin American Journal of Probability and Mathematical Statistics*, vol. 15, no. 2, ALEA, 2018, pp. 1311–34, doi:10.30757/ALEA.v15-49."},"year":"2018","file_date_updated":"2020-07-14T12:47:46Z","author":[{"full_name":"Nejjar, Peter","last_name":"Nejjar","id":"4BF426E2-F248-11E8-B48F-1D18A9856A87","first_name":"Peter"}],"type":"journal_article","publication_identifier":{"issn":["1980-0436"]},"date_published":"2018-10-01T00:00:00Z"},{"pubrep_id":"1040","date_updated":"2021-01-12T08:06:48Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"status":"public","department":[{"_id":"LaEr"}],"supervisor":[{"last_name":"Erdös","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László","orcid":"0000-0001-5366-9603","first_name":"László"}],"title":"Dyson equation and eigenvalue statistics of random matrices","_id":"149","file":[{"file_size":5801709,"date_updated":"2020-07-14T12:44:57Z","creator":"dernst","file_id":"6241","access_level":"open_access","file_name":"2018_thesis_Alt.pdf","content_type":"application/pdf","checksum":"d4dad55a7513f345706aaaba90cb1bb8","date_created":"2019-04-08T13:55:20Z","relation":"main_file"},{"date_updated":"2020-07-14T12:44:57Z","file_size":3802059,"file_name":"2018_thesis_Alt_source.zip","access_level":"closed","creator":"dernst","file_id":"6242","checksum":"d73fcf46300dce74c403f2b491148ab4","content_type":"application/zip","date_created":"2019-04-08T13:55:20Z","relation":"source_file"}],"project":[{"_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804","call_identifier":"FP7","name":"Random matrices, universality and disordered quantum systems"}],"related_material":{"record":[{"id":"1010","status":"public","relation":"part_of_dissertation"},{"relation":"part_of_dissertation","status":"public","id":"1677"},{"id":"550","status":"public","relation":"part_of_dissertation"},{"relation":"part_of_dissertation","id":"566","status":"public"},{"id":"6183","status":"public","relation":"part_of_dissertation"},{"relation":"part_of_dissertation","status":"public","id":"6184"},{"id":"6240","status":"public","relation":"part_of_dissertation"}]},"abstract":[{"text":"The eigenvalue density of many large random matrices is well approximated by a deterministic measure, the self-consistent density of states. In the present work, we show this behaviour for several classes of random matrices. In fact, we establish that, in each of these classes, the self-consistent density of states approximates the eigenvalue density of the random matrix on all scales slightly above the typical eigenvalue spacing. For large classes of random matrices, the self-consistent density of states exhibits several universal features. We prove that, under suitable assumptions, random Gram matrices and Hermitian random matrices with decaying correlations have a 1/3-Hölder continuous self-consistent density of states ρ on R, which is analytic, where it is positive, and has either a square root edge or a cubic root cusp, where it vanishes. We, thus, extend the validity of the corresponding result for Wigner-type matrices from [4, 5, 7]. We show that ρ is determined as the inverse Stieltjes transform of the normalized trace of the unique solution m(z) to the Dyson equation −m(z) −1 = z − a + S[m(z)] on C N×N with the constraint Im m(z) ≥ 0. Here, z lies in the complex upper half-plane, a is a self-adjoint element of C N×N and S is a positivity-preserving operator on C N×N encoding the first two moments of the random matrix. In order to analyze a possible limit of ρ for N → ∞ and address some applications in free probability theory, we also consider the Dyson equation on infinite dimensional von Neumann algebras. We present two applications to random matrices. We first establish that, under certain assumptions, large random matrices with independent entries have a rotationally symmetric self-consistent density of states which is supported on a centered disk in C. Moreover, it is infinitely often differentiable apart from a jump on the boundary of this disk. Second, we show edge universality at all regular (not necessarily extreme) spectral edges for Hermitian random matrices with decaying correlations.","lang":"eng"}],"language":[{"iso":"eng"}],"ddc":["515","519"],"page":"456","doi":"10.15479/AT:ISTA:TH_1040","publisher":"IST Austria","publication_status":"published","date_created":"2018-12-11T11:44:53Z","oa":1,"license":"https://creativecommons.org/licenses/by/4.0/","oa_version":"Published Version","ec_funded":1,"alternative_title":["IST Austria Thesis"],"month":"07","has_accepted_license":"1","day":"12","citation":{"apa":"Alt, J. (2018). *Dyson equation and eigenvalue statistics of random matrices*. IST Austria. https://doi.org/10.15479/AT:ISTA:TH_1040","ista":"Alt J. 2018. Dyson equation and eigenvalue statistics of random matrices. IST Austria.","mla":"Alt, Johannes. *Dyson Equation and Eigenvalue Statistics of Random Matrices*. IST Austria, 2018, doi:10.15479/AT:ISTA:TH_1040.","chicago":"Alt, Johannes. “Dyson Equation and Eigenvalue Statistics of Random Matrices.” IST Austria, 2018. https://doi.org/10.15479/AT:ISTA:TH_1040.","ieee":"J. Alt, “Dyson equation and eigenvalue statistics of random matrices,” IST Austria, 2018.","short":"J. Alt, Dyson Equation and Eigenvalue Statistics of Random Matrices, IST Austria, 2018.","ama":"Alt J. Dyson equation and eigenvalue statistics of random matrices. 2018. doi:10.15479/AT:ISTA:TH_1040"},"type":"dissertation","author":[{"first_name":"Johannes","id":"36D3D8B6-F248-11E8-B48F-1D18A9856A87","last_name":"Alt","full_name":"Alt, Johannes"}],"year":"2018","file_date_updated":"2020-07-14T12:44:57Z","publist_id":"7772","date_published":"2018-07-12T00:00:00Z"},{"month":"05","day":"18","intvolume":" 2018","author":[{"first_name":"László","orcid":"0000-0001-5366-9603","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","last_name":"Erdös"},{"orcid":"0000-0002-2904-1856","first_name":"Dominik J","last_name":"Schröder","id":"408ED176-F248-11E8-B48F-1D18A9856A87","full_name":"Schröder, Dominik J"}],"type":"journal_article","year":"2018","citation":{"chicago":"Erdös, László, and Dominik J Schröder. “Fluctuations of Rectangular Young Diagrams of Interlacing Wigner Eigenvalues.” *International Mathematics Research Notices*. Oxford University Press, 2018. https://doi.org/10.1093/imrn/rnw330.","ieee":"L. Erdös and D. J. Schröder, “Fluctuations of rectangular young diagrams of interlacing wigner eigenvalues,” *International Mathematics Research Notices*, vol. 2018, no. 10. Oxford University Press, pp. 3255–3298, 2018.","short":"L. Erdös, D.J. Schröder, International Mathematics Research Notices 2018 (2018) 3255–3298.","ama":"Erdös L, Schröder DJ. Fluctuations of rectangular young diagrams of interlacing wigner eigenvalues. *International Mathematics Research Notices*. 2018;2018(10):3255-3298. doi:10.1093/imrn/rnw330","apa":"Erdös, L., & Schröder, D. J. (2018). Fluctuations of rectangular young diagrams of interlacing wigner eigenvalues. *International Mathematics Research Notices*. Oxford University Press. https://doi.org/10.1093/imrn/rnw330","ista":"Erdös L, Schröder DJ. 2018. Fluctuations of rectangular young diagrams of interlacing wigner eigenvalues. International Mathematics Research Notices. 2018(10), 3255–3298.","mla":"Erdös, László, and Dominik J. Schröder. “Fluctuations of Rectangular Young Diagrams of Interlacing Wigner Eigenvalues.” *International Mathematics Research Notices*, vol. 2018, no. 10, Oxford University Press, 2018, pp. 3255–98, doi:10.1093/imrn/rnw330."},"publication_identifier":{"issn":["10737928"]},"publist_id":"6383","date_published":"2018-05-18T00:00:00Z","issue":"10","publisher":"Oxford University Press","publication_status":"published","doi":"10.1093/imrn/rnw330","oa":1,"date_created":"2018-12-11T11:49:41Z","oa_version":"Preprint","quality_controlled":"1","ec_funded":1,"_id":"1012","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1608.05163"}],"publication":"International Mathematics Research Notices","title":"Fluctuations of rectangular young diagrams of interlacing wigner eigenvalues","project":[{"name":"Random matrices, universality and disordered quantum systems","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804","call_identifier":"FP7"}],"abstract":[{"text":"We prove a new central limit theorem (CLT) for the difference of linear eigenvalue statistics of a Wigner random matrix H and its minor H and find that the fluctuation is much smaller than the fluctuations of the individual linear statistics, as a consequence of the strong correlation between the eigenvalues of H and H. In particular, our theorem identifies the fluctuation of Kerov's rectangular Young diagrams, defined by the interlacing eigenvalues ofH and H, around their asymptotic shape, the Vershik'Kerov'Logan'Shepp curve. Young diagrams equipped with the Plancherel measure follow the same limiting shape. For this, algebraically motivated, ensemble a CLT has been obtained in Ivanov and Olshanski [20] which is structurally similar to our result but the variance is different, indicating that the analogy between the two models has its limitations. Moreover, our theorem shows that Borodin's result [7] on the convergence of the spectral distribution of Wigner matrices to a Gaussian free field also holds in derivative sense.","lang":"eng"}],"language":[{"iso":"eng"}],"scopus_import":1,"related_material":{"record":[{"relation":"dissertation_contains","status":"public","id":"6179"}]},"page":"3255-3298","date_updated":"2021-01-12T08:06:34Z","volume":2018,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","external_id":{"arxiv":["1608.05163"]},"department":[{"_id":"LaEr"}],"status":"public"},{"language":[{"iso":"eng"}],"abstract":[{"lang":"eng","text":"Let S be a positivity-preserving symmetric linear operator acting on bounded functions. The nonlinear equation -1/m=z+Sm with a parameter z in the complex upper half-plane ℍ has a unique solution m with values in ℍ. We show that the z-dependence of this solution can be represented as the Stieltjes transforms of a family of probability measures v on ℝ. Under suitable conditions on S, we show that v has a real analytic density apart from finitely many algebraic singularities of degree at most 3. Our motivation comes from large random matrices. The solution m determines the density of eigenvalues of two prominent matrix ensembles: (i) matrices with centered independent entries whose variances are given by S and (ii) matrices with correlated entries with a translation-invariant correlation structure. Our analysis shows that the limiting eigenvalue density has only square root singularities or cubic root cusps; no other singularities occur."}],"scopus_import":1,"page":"1672 - 1705","main_file_link":[{"url":"https://arxiv.org/abs/1512.03703","open_access":"1"}],"_id":"721","title":"Singularities of solutions to quadratic vector equations on the complex upper half plane","publication":"Communications on Pure and Applied Mathematics","project":[{"call_identifier":"FP7","grant_number":"338804","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","name":"Random matrices, universality and disordered quantum systems"}],"department":[{"_id":"LaEr"}],"status":"public","date_updated":"2021-01-12T08:12:24Z","user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","volume":70,"publication_identifier":{"issn":["00103640"]},"date_published":"2017-09-01T00:00:00Z","publist_id":"6959","day":"01","month":"09","intvolume":" 70","year":"2017","author":[{"full_name":"Ajanki, Oskari H","last_name":"Ajanki","id":"36F2FB7E-F248-11E8-B48F-1D18A9856A87","first_name":"Oskari H"},{"first_name":"Torben H","orcid":"0000-0002-4821-3297","full_name":"Krüger, Torben H","last_name":"Krüger","id":"3020C786-F248-11E8-B48F-1D18A9856A87"},{"orcid":"0000-0001-5366-9603","first_name":"László","last_name":"Erdös","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","full_name":"Erdös, László"}],"type":"journal_article","citation":{"mla":"Ajanki, Oskari H., et al. “Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half Plane.” *Communications on Pure and Applied Mathematics*, vol. 70, no. 9, Wiley-Blackwell, 2017, pp. 1672–705, doi:10.1002/cpa.21639.","ista":"Ajanki OH, Krüger TH, Erdös L. 2017. Singularities of solutions to quadratic vector equations on the complex upper half plane. Communications on Pure and Applied Mathematics. 70(9), 1672–1705.","apa":"Ajanki, O. H., Krüger, T. H., & Erdös, L. (2017). Singularities of solutions to quadratic vector equations on the complex upper half plane. *Communications on Pure and Applied Mathematics*. Wiley-Blackwell. https://doi.org/10.1002/cpa.21639","short":"O.H. Ajanki, T.H. Krüger, L. Erdös, Communications on Pure and Applied Mathematics 70 (2017) 1672–1705.","ieee":"O. H. Ajanki, T. H. Krüger, and L. Erdös, “Singularities of solutions to quadratic vector equations on the complex upper half plane,” *Communications on Pure and Applied Mathematics*, vol. 70, no. 9. Wiley-Blackwell, pp. 1672–1705, 2017.","ama":"Ajanki OH, Krüger TH, Erdös L. Singularities of solutions to quadratic vector equations on the complex upper half plane. *Communications on Pure and Applied Mathematics*. 2017;70(9):1672-1705. doi:10.1002/cpa.21639","chicago":"Ajanki, Oskari H, Torben H Krüger, and László Erdös. “Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half Plane.” *Communications on Pure and Applied Mathematics*. Wiley-Blackwell, 2017. https://doi.org/10.1002/cpa.21639."},"oa_version":"Submitted Version","ec_funded":1,"quality_controlled":"1","issue":"9","date_created":"2018-12-11T11:48:08Z","oa":1,"publication_status":"published","doi":"10.1002/cpa.21639","publisher":"Wiley-Blackwell"},{"project":[{"name":"Random matrices, universality and disordered quantum systems","_id":"258DCDE6-B435-11E9-9278-68D0E5697425","grant_number":"338804","call_identifier":"FP7"}],"title":"Convergence rate for spectral distribution of addition of random matrices","publication":"Advances in Mathematics","main_file_link":[{"url":"https://arxiv.org/abs/1606.03076","open_access":"1"}],"_id":"733","acknowledgement":"Partially supported by ERC Advanced Grant RANMAT No. 338804, Hong Kong RGC grant ECS 26301517, and the Göran Gustafsson Foundation","page":"251 - 291","scopus_import":1,"language":[{"iso":"eng"}],"abstract":[{"text":"Let A and B be two N by N deterministic Hermitian matrices and let U be an N by N Haar distributed unitary matrix. It is well known that the spectral distribution of the sum H = A + UBU∗ converges weakly to the free additive convolution of the spectral distributions of A and B, as N tends to infinity. We establish the optimal convergence rate in the bulk of the spectrum.","lang":"eng"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","volume":319,"date_updated":"2021-01-12T08:13:07Z","department":[{"_id":"LaEr"}],"status":"public","citation":{"ista":"Bao Z, Erdös L, Schnelli K. 2017. Convergence rate for spectral distribution of addition of random matrices. Advances in Mathematics. 319, 251–291.","apa":"Bao, Z., Erdös, L., & Schnelli, K. (2017). Convergence rate for spectral distribution of addition of random matrices. *Advances in Mathematics*. Academic Press. https://doi.org/10.1016/j.aim.2017.08.028","mla":"Bao, Zhigang, et al. “Convergence Rate for Spectral Distribution of Addition of Random Matrices.” *Advances in Mathematics*, vol. 319, Academic Press, 2017, pp. 251–91, doi:10.1016/j.aim.2017.08.028.","chicago":"Bao, Zhigang, László Erdös, and Kevin Schnelli. “Convergence Rate for Spectral Distribution of Addition of Random Matrices.” *Advances in Mathematics*. Academic Press, 2017. https://doi.org/10.1016/j.aim.2017.08.028.","short":"Z. Bao, L. Erdös, K. Schnelli, Advances in Mathematics 319 (2017) 251–291.","ieee":"Z. Bao, L. Erdös, and K. Schnelli, “Convergence rate for spectral distribution of addition of random matrices,” *Advances in Mathematics*, vol. 319. Academic Press, pp. 251–291, 2017.","ama":"Bao Z, Erdös L, Schnelli K. Convergence rate for spectral distribution of addition of random matrices. *Advances in Mathematics*. 2017;319:251-291. doi:10.1016/j.aim.2017.08.028"},"year":"2017","author":[{"full_name":"Bao, Zhigang","id":"442E6A6C-F248-11E8-B48F-1D18A9856A87","last_name":"Bao","first_name":"Zhigang","orcid":"0000-0003-3036-1475"},{"first_name":"László","orcid":"0000-0001-5366-9603","full_name":"Erdös, László","id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","last_name":"Erdös"},{"full_name":"Schnelli, Kevin","id":"434AD0AE-F248-11E8-B48F-1D18A9856A87","last_name":"Schnelli","first_name":"Kevin","orcid":"0000-0003-0954-3231"}],"type":"journal_article","intvolume":" 319","day":"15","month":"10","date_published":"2017-10-15T00:00:00Z","publist_id":"6935","oa":1,"date_created":"2018-12-11T11:48:13Z","publication_status":"published","publisher":"Academic Press","doi":"10.1016/j.aim.2017.08.028","quality_controlled":"1","ec_funded":1,"oa_version":"Submitted Version"}]