[{"acknowledgement":"The authors are grateful to Milán Mosonyi for fruitful discussions on the topic, and to the anonymous referee for his/her comments and suggestions.\r\nJ. Pitrik was supported by the Hungarian Academy of Sciences Lendület-Momentum Grant for Quantum Information Theory, No. 96 141, and by Hungarian National Research, Development and Innovation Office (NKFIH) via grants no. K119442, no. K124152, and no. KH129601. D. Virosztek was supported by the ISTFELLOW program of the Institute of Science and Technology Austria (project code IC1027FELL01), by the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie Grant Agreement No. 846294, and partially supported by the Hungarian National Research, Development and Innovation Office (NKFIH) via grants no. K124152, and no. KH129601.","publication_identifier":{"issn":["0024-3795"]},"date_published":"2021-01-15T00:00:00Z","publication_status":"published","type":"journal_article","title":"A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means","date_created":"2020-09-11T08:35:50Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_updated":"2021-01-11T16:28:42Z","language":[{"iso":"eng"}],"year":"2021","ec_funded":1,"main_file_link":[{"url":"https://arxiv.org/abs/2002.11678","open_access":"1"}],"page":"203-217","abstract":[{"text":"It is well known that special Kubo-Ando operator means admit divergence center interpretations, moreover, they are also mean squared error estimators for certain metrics on positive definite operators. In this paper we give a divergence center interpretation for every symmetric Kubo-Ando mean. This characterization of the symmetric means naturally leads to a definition of weighted and multivariate versions of a large class of symmetric Kubo-Ando means. We study elementary properties of these weighted multivariate means, and note in particular that in the special case of the geometric mean we recover the weighted A#H-mean introduced by Kim, Lawson, and Lim.","lang":"eng"}],"external_id":{"arxiv":["2002.11678"]},"volume":609,"oa":1,"author":[{"first_name":"József","last_name":"Pitrik","full_name":"Pitrik, József"},{"orcid":"0000-0003-1109-5511","first_name":"Daniel","full_name":"Virosztek, Daniel","last_name":"Virosztek","id":"48DB45DA-F248-11E8-B48F-1D18A9856A87"}],"department":[{"_id":"LaEr"}],"article_type":"original","day":"15","month":"01","keyword":["Kubo-Ando mean","weighted multivariate mean","barycenter"],"article_processing_charge":"No","quality_controlled":"1","intvolume":" 609","citation":{"apa":"Pitrik, J., & Virosztek, D. (2021). A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means. *Linear Algebra and Its Applications*. Elsevier. https://doi.org/10.1016/j.laa.2020.09.007","ieee":"J. Pitrik and D. Virosztek, “A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means,” *Linear Algebra and its Applications*, vol. 609. Elsevier, pp. 203–217, 2021.","short":"J. Pitrik, D. Virosztek, Linear Algebra and Its Applications 609 (2021) 203–217.","chicago":"Pitrik, József, and Daniel Virosztek. “A Divergence Center Interpretation of General Symmetric Kubo-Ando Means, and Related Weighted Multivariate Operator Means.” *Linear Algebra and Its Applications*. Elsevier, 2021. https://doi.org/10.1016/j.laa.2020.09.007.","mla":"Pitrik, József, and Daniel Virosztek. “A Divergence Center Interpretation of General Symmetric Kubo-Ando Means, and Related Weighted Multivariate Operator Means.” *Linear Algebra and Its Applications*, vol. 609, Elsevier, 2021, pp. 203–17, doi:10.1016/j.laa.2020.09.007.","ama":"Pitrik J, Virosztek D. A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means. *Linear Algebra and its Applications*. 2021;609:203-217. doi:10.1016/j.laa.2020.09.007","ista":"Pitrik J, Virosztek D. 2021. A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means. Linear Algebra and its Applications. 609, 203–217."},"project":[{"grant_number":"846294","_id":"26A455A6-B435-11E9-9278-68D0E5697425","name":"Geometric study of Wasserstein spaces and free probability","call_identifier":"H2020"},{"grant_number":"291734","name":"International IST Postdoc Fellowship Programme","_id":"25681D80-B435-11E9-9278-68D0E5697425","call_identifier":"FP7"}],"doi":"10.1016/j.laa.2020.09.007","publisher":"Elsevier","oa_version":"Preprint","status":"public","_id":"8373","publication":"Linear Algebra and its Applications"},{"citation":{"mla":"Cipolloni, Giorgio. *Fluctuations in the Spectrum of Random Matrices*. IST Austria, 2021, doi:10.15479/AT:ISTA:9022.","chicago":"Cipolloni, Giorgio. “Fluctuations in the Spectrum of Random Matrices.” IST Austria, 2021. https://doi.org/10.15479/AT:ISTA:9022.","ista":"Cipolloni G. 2021. Fluctuations in the spectrum of random matrices. IST Austria.","ama":"Cipolloni G. Fluctuations in the spectrum of random matrices. 2021. doi:10.15479/AT:ISTA:9022","apa":"Cipolloni, G. (2021). *Fluctuations in the spectrum of random matrices*. IST Austria. https://doi.org/10.15479/AT:ISTA:9022","short":"G. Cipolloni, Fluctuations in the Spectrum of Random Matrices, IST Austria, 2021.","ieee":"G. Cipolloni, “Fluctuations in the spectrum of random matrices,” IST Austria, 2021."},"has_accepted_license":"1","article_processing_charge":"No","day":"25","month":"01","department":[{"_id":"GradSch"},{"_id":"LaEr"}],"_id":"9022","status":"public","file":[{"file_size":4127796,"checksum":"5a93658a5f19478372523ee232887e2b","success":1,"file_id":"9043","access_level":"open_access","date_updated":"2021-01-25T14:19:03Z","date_created":"2021-01-25T14:19:03Z","creator":"gcipollo","content_type":"application/pdf","file_name":"thesis.pdf","relation":"main_file"},{"date_updated":"2021-01-25T14:19:10Z","file_id":"9044","access_level":"closed","checksum":"e8270eddfe6a988e92a53c88d1d19b8c","file_size":12775206,"relation":"source_file","file_name":"Thesis_files.zip","content_type":"application/zip","creator":"gcipollo","date_created":"2021-01-25T14:19:10Z"}],"publisher":"IST Austria","oa_version":"Published Version","supervisor":[{"id":"4DBD5372-F248-11E8-B48F-1D18A9856A87","last_name":"Erdös","full_name":"Erdös, László","first_name":"László","orcid":"0000-0001-5366-9603"}],"project":[{"call_identifier":"H2020","grant_number":"665385","name":"International IST Doctoral Program","_id":"2564DBCA-B435-11E9-9278-68D0E5697425"},{"call_identifier":"FP7","grant_number":"338804","name":"Random matrices, universality and disordered quantum systems","_id":"258DCDE6-B435-11E9-9278-68D0E5697425"}],"doi":"10.15479/AT:ISTA:9022","date_created":"2021-01-21T18:16:54Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","ddc":["510"],"title":"Fluctuations in the spectrum of random matrices","type":"dissertation","publication_status":"published","date_published":"2021-01-25T00:00:00Z","alternative_title":["IST Austria Thesis"],"publication_identifier":{"eissn":["2663-337X"]},"file_date_updated":"2021-01-25T14:19:10Z","acknowledgement":"I gratefully acknowledge the financial support from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 665385 and my advisor’s ERC Advanced Grant No. 338804.","author":[{"first_name":"Giorgio","last_name":"Cipolloni","id":"42198EFA-F248-11E8-B48F-1D18A9856A87","full_name":"Cipolloni, Giorgio"}],"oa":1,"abstract":[{"lang":"eng","text":"In the first part of the thesis we consider Hermitian random matrices. Firstly, we consider sample covariance matrices XX∗ with X having independent identically distributed (i.i.d.) centred entries. We prove a Central Limit Theorem for differences of linear statistics of XX∗ and its minor after removing the first column of X. Secondly, we consider Wigner-type matrices and prove that the eigenvalue statistics near cusp singularities of the limiting density of states are universal and that they form a Pearcey process. Since the limiting eigenvalue distribution admits only square root (edge) and cubic root (cusp) singularities, this concludes the third and last remaining case of the Wigner-Dyson-Mehta universality conjecture. The main technical ingredients are an optimal local law at the cusp, and the proof of the fast relaxation to equilibrium of the Dyson Brownian motion in the cusp regime.\r\nIn the second part we consider non-Hermitian matrices X with centred i.i.d. entries. We normalise the entries of X to have variance N −1. It is well known that the empirical eigenvalue density converges to the uniform distribution on the unit disk (circular law). In the first project, we prove universality of the local eigenvalue statistics close to the edge of the spectrum. This is the non-Hermitian analogue of the TracyWidom universality at the Hermitian edge. Technically we analyse the evolution of the spectral distribution of X along the Ornstein-Uhlenbeck flow for very long time\r\n(up to t = +∞). In the second project, we consider linear statistics of eigenvalues for macroscopic test functions f in the Sobolev space H2+ϵ and prove their convergence to the projection of the Gaussian Free Field on the unit disk. We prove this result for non-Hermitian matrices with real or complex entries. The main technical ingredients are: (i) local law for products of two resolvents at different spectral parameters, (ii) analysis of correlated Dyson Brownian motions.\r\nIn the third and final part we discuss the mathematically rigorous application of supersymmetric techniques (SUSY ) to give a lower tail estimate of the lowest singular value of X − z, with z ∈ C. More precisely, we use superbosonisation formula to give an integral representation of the resolvent of (X − z)(X − z)∗ which reduces to two and three contour integrals in the complex and real case, respectively. The rigorous analysis of these integrals is quite challenging since simple saddle point analysis cannot be applied (the main contribution comes from a non-trivial manifold). Our result\r\nimproves classical smoothing inequalities in the regime |z| ≈ 1; this result is essential to prove edge universality for i.i.d. non-Hermitian matrices."}],"page":"380","ec_funded":1,"language":[{"iso":"eng"}],"year":"2021","date_updated":"2021-02-04T14:51:51Z"},{"acknowledgement":"The research of L.-P. A. is supported in part by the grant NSF CAREER DMS-1653602. G. D. gratefully acknowledges support from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 754411. The research of L. H. is supported in part by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through Project-ID 233630050 -TRR 146, Project-ID 443891315 within SPP 2265 and Project-ID 446173099.","department":[{"_id":"LaEr"}],"article_processing_charge":"No","date_published":"2021-03-08T00:00:00Z","month":"03","day":"08","title":"Maxima of a random model of the Riemann zeta function over intervals of varying length","type":"preprint","publication_status":"submitted","citation":{"apa":"Arguin, L.-P., Dubach, G., & Hartung, L. (n.d.). Maxima of a random model of the Riemann zeta function over intervals of varying length. *arXiv*.","ieee":"L.-P. Arguin, G. Dubach, and L. Hartung, “Maxima of a random model of the Riemann zeta function over intervals of varying length,” *arXiv*. .","short":"L.-P. Arguin, G. Dubach, L. Hartung, ArXiv (n.d.).","chicago":"Arguin, Louis-Pierre, Guillaume Dubach, and Lisa Hartung. “Maxima of a Random Model of the Riemann Zeta Function over Intervals of Varying Length.” *ArXiv*, n.d.","mla":"Arguin, Louis-Pierre, et al. “Maxima of a Random Model of the Riemann Zeta Function over Intervals of Varying Length.” *ArXiv*, 2103.04817.","ista":"Arguin L-P, Dubach G, Hartung L. Maxima of a random model of the Riemann zeta function over intervals of varying length. arXiv, 2103.04817.","ama":"Arguin L-P, Dubach G, Hartung L. Maxima of a random model of the Riemann zeta function over intervals of varying length. *arXiv*."},"date_created":"2021-03-09T11:08:15Z","article_number":"2103.04817","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2103.04817"}],"ec_funded":1,"year":"2021","language":[{"iso":"eng"}],"date_updated":"2021-03-09T12:07:57Z","external_id":{"arxiv":["2103.04817"]},"abstract":[{"text":"We consider a model of the Riemann zeta function on the critical axis and study its maximum over intervals of length (log T)θ, where θ is either fixed or tends to zero at a suitable rate.\r\nIt is shown that the deterministic level of the maximum interpolates smoothly between the ones\r\nof log-correlated variables and of i.i.d. random variables, exhibiting a smooth transition ‘from\r\n3/4 to 1/4’ in the second order. This provides a natural context where extreme value statistics of\r\nlog-correlated variables with time-dependent variance and rate occur. A key ingredient of the\r\nproof is a precise upper tail tightness estimate for the maximum of the model on intervals of\r\nsize one, that includes a Gaussian correction. This correction is expected to be present for the\r\nRiemann zeta function and pertains to the question of the correct order of the maximum of\r\nthe zeta function in large intervals.","lang":"eng"}],"oa_version":"Preprint","project":[{"call_identifier":"H2020","grant_number":"754411","_id":"260C2330-B435-11E9-9278-68D0E5697425","name":"ISTplus - Postdoctoral Fellowships"}],"publication":"arXiv","_id":"9230","author":[{"first_name":"Louis-Pierre","full_name":"Arguin, Louis-Pierre","last_name":"Arguin"},{"full_name":"Dubach, Guillaume","id":"D5C6A458-10C4-11EA-ABF4-A4B43DDC885E","last_name":"Dubach","first_name":"Guillaume","orcid":"0000-0001-6892-8137"},{"first_name":"Lisa","full_name":"Hartung, Lisa","last_name":"Hartung"}],"status":"public","oa":1},{"oa":1,"_id":"9281","publication":"arXiv","author":[{"last_name":"Dubach","id":"D5C6A458-10C4-11EA-ABF4-A4B43DDC885E","full_name":"Dubach, Guillaume","first_name":"Guillaume","orcid":"0000-0001-6892-8137"},{"orcid":"0000-0003-1548-0177","first_name":"Fabian","last_name":"Mühlböck","id":"6395C5F6-89DF-11E9-9C97-6BDFE5697425","full_name":"Mühlböck, Fabian"}],"status":"public","project":[{"call_identifier":"H2020","grant_number":"754411","name":"ISTplus - Postdoctoral Fellowships","_id":"260C2330-B435-11E9-9278-68D0E5697425"}],"oa_version":"Preprint","abstract":[{"text":"We comment on two formal proofs of Fermat's sum of two squares theorem, written using the Mathematical Components libraries of the Coq proof assistant. The first one follows Zagier's celebrated one-sentence proof; the second follows David Christopher's recent new proof relying on partition-theoretic arguments. Both formal proofs rely on a general property of involutions of finite sets, of independent interest. The proof technique consists for the most part of automating recurrent tasks (such as case distinctions and computations on natural numbers) via ad hoc tactics.","lang":"eng"}],"external_id":{"arxiv":["2103.11389"]},"language":[{"iso":"eng"}],"year":"2021","date_updated":"2021-03-23T09:04:32Z","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2103.11389"}],"ec_funded":1,"citation":{"apa":"Dubach, G., & Mühlböck, F. (n.d.). Formal verification of Zagier’s one-sentence proof. *arXiv*.","ieee":"G. Dubach and F. Mühlböck, “Formal verification of Zagier’s one-sentence proof,” *arXiv*. .","short":"G. Dubach, F. Mühlböck, ArXiv (n.d.).","chicago":"Dubach, Guillaume, and Fabian Mühlböck. “Formal Verification of Zagier’s One-Sentence Proof.” *ArXiv*, n.d.","mla":"Dubach, Guillaume, and Fabian Mühlböck. “Formal Verification of Zagier’s One-Sentence Proof.” *ArXiv*, 2103.11389.","ista":"Dubach G, Mühlböck F. Formal verification of Zagier’s one-sentence proof. arXiv, 2103.11389.","ama":"Dubach G, Mühlböck F. Formal verification of Zagier’s one-sentence proof. *arXiv*."},"article_number":"2103.11389","date_created":"2021-03-23T05:38:48Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","type":"preprint","publication_status":"draft","title":"Formal verification of Zagier's one-sentence proof","month":"03","day":"21","date_published":"2021-03-21T00:00:00Z","article_processing_charge":"No","department":[{"_id":"LaEr"},{"_id":"ToHe"}]},{"date_published":"2021-03-26T00:00:00Z","acknowledgement":"D. Virosztek was supported by the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 846294, and partially supported by the Hungarian National Research, Development and Innovation Office (NKFIH) via grants no. K124152, and no. KH129601.","publication_identifier":{"issn":["0001-8708"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2021-01-22T17:55:17Z","publication_status":"published","type":"journal_article","title":"The metric property of the quantum Jensen-Shannon divergence","issue":"3","abstract":[{"text":"In this short note, we prove that the square root of the quantum Jensen-Shannon divergence is a true metric on the cone of positive matrices, and hence in particular on the quantum state space.","lang":"eng"}],"external_id":{"arxiv":["1910.10447"]},"date_updated":"2021-04-12T14:01:57Z","language":[{"iso":"eng"}],"year":"2021","ec_funded":1,"main_file_link":[{"url":"https://arxiv.org/abs/1910.10447","open_access":"1"}],"volume":380,"oa":1,"author":[{"id":"48DB45DA-F248-11E8-B48F-1D18A9856A87","last_name":"Virosztek","full_name":"Virosztek, Daniel","orcid":"0000-0003-1109-5511","first_name":"Daniel"}],"day":"26","month":"03","keyword":["General Mathematics"],"article_processing_charge":"No","department":[{"_id":"LaEr"}],"article_type":"original","intvolume":" 380","article_number":"107595","citation":{"apa":"Virosztek, D. (2021). The metric property of the quantum Jensen-Shannon divergence. *Advances in Mathematics*. Elsevier. https://doi.org/10.1016/j.aim.2021.107595","short":"D. Virosztek, Advances in Mathematics 380 (2021).","ieee":"D. Virosztek, “The metric property of the quantum Jensen-Shannon divergence,” *Advances in Mathematics*, vol. 380, no. 3. Elsevier, 2021.","mla":"Virosztek, Daniel. “The Metric Property of the Quantum Jensen-Shannon Divergence.” *Advances in Mathematics*, vol. 380, no. 3, 107595, Elsevier, 2021, doi:10.1016/j.aim.2021.107595.","chicago":"Virosztek, Daniel. “The Metric Property of the Quantum Jensen-Shannon Divergence.” *Advances in Mathematics*. Elsevier, 2021. https://doi.org/10.1016/j.aim.2021.107595.","ama":"Virosztek D. The metric property of the quantum Jensen-Shannon divergence. *Advances in Mathematics*. 2021;380(3). doi:10.1016/j.aim.2021.107595","ista":"Virosztek D. 2021. The metric property of the quantum Jensen-Shannon divergence. Advances in Mathematics. 380(3), 107595."},"quality_controlled":"1","status":"public","_id":"9036","publication":"Advances in Mathematics","doi":"10.1016/j.aim.2021.107595","project":[{"grant_number":"846294","name":"Geometric study of Wasserstein spaces and free probability","_id":"26A455A6-B435-11E9-9278-68D0E5697425","call_identifier":"H2020"}],"publisher":"Elsevier","oa_version":"Preprint"}]