{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,17]],"date-time":"2026-08-17T15:24:31Z","timestamp":1786980271120,"version":"build-2736575974"},"reference-count":33,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2022,3,14]],"date-time":"2022-03-14T00:00:00Z","timestamp":1647216000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2022,3,14]],"date-time":"2022-03-14T00:00:00Z","timestamp":1647216000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Combinatorica"],"published-print":{"date-parts":[[2022,8]]},"DOI":"10.1007\/s00493-020-4657-0","type":"journal-article","created":{"date-parts":[[2022,3,14]],"date-time":"2022-03-14T09:17:03Z","timestamp":1647249423000},"page":"491-527","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["The Characteristic Polynomial of a Random Matrix"],"prefix":"10.1007","volume":"42","author":[{"given":"Sean","family":"Eberhard","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2022,3,14]]},"reference":[{"key":"4657_CR1","doi-asserted-by":"publisher","first-page":"1620","DOI":"10.1214\/aop\/1019160500","volume":"28","author":"R Arratia","year":"2000","unstructured":"R. Arratia, A. D. Barbour and S. Tavar\u00e9: Limits of logarithmic combinatorial structures, Ann. Probab. 28 (2000), 1620\u20131644.","journal-title":"Ann. Probab."},{"key":"4657_CR2","doi-asserted-by":"crossref","unstructured":"R. Arratia, A. D. Barbour and S. Tavar\u00e9: Logarithmic combinatorial structures: a probabilistic approach, EMS Monographs in Mathematics. European Mathematical Society (EMS), Z\u00fcrich, 2003.","DOI":"10.4171\/000"},{"key":"4657_CR3","unstructured":"L. Bary-Soroker, D. Koukoulopoulos and G. Kozma: Irreducibility of random polynomials: general measures, arXiv:2007.14567, July 2020."},{"key":"4657_CR4","doi-asserted-by":"publisher","first-page":"579","DOI":"10.1215\/00127094-2019-0047","volume":"169","author":"L Bary-Soroker","year":"2020","unstructured":"L. Bary-Soroker and G. Kozma: Irreducible polynomials of bounded height, Duke Math. J. 169 (2020), 579\u2013598.","journal-title":"Duke Math. J."},{"key":"4657_CR5","doi-asserted-by":"publisher","first-page":"559","DOI":"10.1016\/j.jfa.2009.04.016","volume":"258","author":"J Bourgain","year":"2010","unstructured":"J. Bourgain, V. H. Vu and Ph. M. Wood: On the singularity probability of discrete random matrices, J. Funct. Anal. 258 (2010), 559\u2013603.","journal-title":"J. Funct. Anal."},{"key":"4657_CR6","doi-asserted-by":"publisher","first-page":"195","DOI":"10.4310\/ACTA.2019.v223.n2.a1","volume":"223","author":"E Breuillard","year":"2019","unstructured":"E. Breuillard and P. P. Varj\u00fa: Irreducibility of random polynomials of large degree, Acta Math. 223 (2019), 195\u2013249.","journal-title":"Acta Math."},{"key":"4657_CR7","unstructured":"M. Campos, M. Jenssen, M. Michelen and J. Sahasrabudhe: The singularity probability of a random symmetric matrix is exponentially small, arXiv:2105.11384, May 2021."},{"key":"4657_CR8","unstructured":"M. Campos, L. Mattos, R. Morris and N. Morrison: On the singularity of random symmetric matrices, 2019."},{"key":"4657_CR9","doi-asserted-by":"publisher","first-page":"395","DOI":"10.1215\/S0012-7094-06-13527-5","volume":"135","author":"K P Costello","year":"2006","unstructured":"K. P. Costello, T. Tao and V. Vu: Random symmetric matrices are almost surely nonsingular, Duke Math. J. 135 (2006), 395\u2013413.","journal-title":"Duke Math. J."},{"key":"4657_CR10","volume-title":"Permutation groups, volume 163 of Graduate Texts in Mathematics","author":"J D Dixon","year":"1996","unstructured":"J. D. Dixon and B. Mortimer: Permutation groups, volume 163 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1996."},{"key":"4657_CR11","doi-asserted-by":"publisher","first-page":"1573","DOI":"10.1215\/00127094-0000007X","volume":"166","author":"S Eberhard","year":"2017","unstructured":"S. Eberhard, K. Ford and B. Green: Invariable generation of the symmetric group, Duke Math. J. 166 (2017), 1573\u20131590.","journal-title":"Duke Math. J."},{"key":"4657_CR12","first-page":"29","volume":"7e22","author":"A Ferber","year":"2019","unstructured":"A. Ferber and V. Jain: Singularity of random symmetric matrices\u2014a combinatorial approach to improved bounds, Forum Math. Sigma, 7:e22, 29, 2019.","journal-title":"Forum Math. Sigma"},{"key":"4657_CR13","doi-asserted-by":"crossref","unstructured":"A. Ferber, V. Jain, A. Sah and M. Sawhney: Random symmetric matrices: rank distribution and irreducibility of the characteristic polynomial, arXiv:2106.04049, June 2021.","DOI":"10.1017\/S0305004122000226"},{"key":"4657_CR14","doi-asserted-by":"publisher","first-page":"507","DOI":"10.1017\/S0305004100071796","volume":"114","author":"J C Hansen","year":"1993","unstructured":"J. C. Hansen and E. Schmutz: How random is the characteristic polynomial of a random matrix? Math. Proc. Cambridge Philos. Soc. 114 (1993), 507\u2013515.","journal-title":"Math. Proc. Cambridge Philos. Soc."},{"key":"4657_CR15","unstructured":"R. A. Horn and Ch. R. Johnson: Matrix analysis, Cambridge University Press, Cambridge, second edition, 2013."},{"key":"4657_CR16","first-page":"7","volume":"2","author":"J Koml\u00f3s","year":"1967","unstructured":"J. Koml\u00f3s: On the determinant of (0,1) matrices, Studia Sci. Math. Hungar. 2 (1967), 7\u201321.","journal-title":"Studia Sci. Math. Hungar."},{"key":"4657_CR17","doi-asserted-by":"crossref","unstructured":"K. Luh, S. Meehan and H. H. Nguyen: Some new results in random matrices over finite fields, arXiv:1907.02575, July 2019.","DOI":"10.1112\/jlms.12405"},{"key":"4657_CR18","doi-asserted-by":"crossref","unstructured":"K. Luh and S. O\u2019Rourke: Eigenvectors and controllability of non-Hermitian random matrices and directed graphs, arXiv:2004.10543, April 2020.","DOI":"10.1214\/21-EJP588"},{"key":"4657_CR19","unstructured":"K. Maples: Singularity of random matrices over finite fields, 2010."},{"key":"4657_CR20","doi-asserted-by":"crossref","unstructured":"P. M. Neumann: Helmut Wielandt on permutation groups, in: Helmut Wielandt Mathematische Werke\/Mathematical Works, Volume 1, Group Theory, 3\u201320, DE GRUYTER.","DOI":"10.1515\/9783110863383.3"},{"key":"4657_CR21","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1017\/S0963548319000348","volume":"29","author":"H H Nguyen","year":"2020","unstructured":"H. H. Nguyen and E. Paquette: Surjectivity of near-square random matrices, Combin. Probab. Comput. 29 (2020), 267\u2013292.","journal-title":"Combin. Probab. Comput."},{"key":"4657_CR22","first-page":"317","volume":"39","author":"A M Odlyzko","year":"1993","unstructured":"A. M. Odlyzko and B. Poonen: Zeros of polynomials with 0,1 coefficients, Enseign. Math. 39 (1993), 317\u2013348.","journal-title":"Enseign. Math."},{"key":"4657_CR23","doi-asserted-by":"publisher","first-page":"1076","DOI":"10.1007\/s10959-018-0839-8","volume":"32","author":"S O\u2019Rourke","year":"2019","unstructured":"S. O\u2019Rourke and Ph. M. Wood: Low-degree factors of random polynomials, J. Theoret. Probab. 32 (2019), 1076\u20131104.","journal-title":"J. Theoret. Probab."},{"key":"4657_CR24","doi-asserted-by":"publisher","first-page":"409","DOI":"10.1002\/rsa.20632","volume":"49","author":"R Pemantle","year":"2016","unstructured":"R. Pemantle, Y. Peres and I. Rivin: Four random permutations conjugated by an adversary generate Sn with high probability, Random Structures Algorithms 49 (2016), 409\u2013428.","journal-title":"Random Structures Algorithms"},{"key":"4657_CR25","doi-asserted-by":"publisher","first-page":"409","DOI":"10.1002\/rsa.20632","volume":"49","author":"R Pemantle","year":"2016","unstructured":"R. Pemantle, Y. Peres and I. Rivin: Four random permutations conjugated by an adversary generate Sn with high probability, Random Structures Algorithms 49 (2016), 409\u2013428.","journal-title":"Random Structures Algorithms"},{"key":"4657_CR26","doi-asserted-by":"publisher","first-page":"324","DOI":"10.1215\/ijm\/1255629830","volume":"5","author":"I Reiner","year":"1961","unstructured":"I. Reiner: On the number of matrices with given characteristic polynomial, Illinois J. Math. 5 (1961), 324\u2013329.","journal-title":"Illinois J. Math."},{"key":"4657_CR27","doi-asserted-by":"publisher","first-page":"525","DOI":"10.1080\/15326349408807308","volume":"10","author":"J W Silverstein","year":"1994","unstructured":"J. W. Silverstein: The spectral radii and norms of large-dimensional non-central random matrices, Comm. Statist. Stochastic Models 10 (1994), 525\u2013532.","journal-title":"Comm. Statist. Stochastic Models"},{"key":"4657_CR28","doi-asserted-by":"publisher","first-page":"539","DOI":"10.1007\/s00493-016-3363-4","volume":"37","author":"T Tao","year":"2017","unstructured":"T. Tao and V. Vu: Random matrices have simple spectrum, Combinatorica 37 (2017), 539\u2013553.","journal-title":"Combinatorica"},{"key":"4657_CR29","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1002\/rsa.20109","volume":"28","author":"T Tao","year":"2005","unstructured":"T. Tao and V. H. Vu: On random \u00b11 matrices: Singularity and determinant, Random Structures and Algorithms 28 (2005), 1\u201323.","journal-title":"Random Structures and Algorithms"},{"key":"4657_CR30","volume-title":"Additive combinatorics, volume 105 of Cambridge Studies in Advanced Mathematics","author":"T Tao","year":"2010","unstructured":"T. Tao and V. H. Vu: Additive combinatorics, volume 105 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2010."},{"key":"4657_CR31","doi-asserted-by":"publisher","first-page":"593","DOI":"10.4007\/annals.2020.191.2.6","volume":"191","author":"K Tikhomirov","year":"2020","unstructured":"K. Tikhomirov: Singularity of random Bernoulli matrices, Ann. of Math. 191 (2020), 593\u2013634.","journal-title":"Ann. of Math."},{"key":"4657_CR32","doi-asserted-by":"crossref","unstructured":"R. Vershynin: High-dimensional probability, volume 47 of Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 2018. An introduction with applications in data science, With a foreword by Sara van de Geer.","DOI":"10.1017\/9781108231596"},{"key":"4657_CR33","doi-asserted-by":"crossref","unstructured":"V. Vu: Recent progress in combinatorial random matrix theory, arXiv:2005.02797, May 2020.","DOI":"10.1214\/20-PS346"}],"container-title":["Combinatorica"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00493-020-4657-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00493-020-4657-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00493-020-4657-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,10,3]],"date-time":"2022-10-03T11:05:16Z","timestamp":1664795116000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00493-020-4657-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,3,14]]},"references-count":33,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2022,8]]}},"alternative-id":["4657"],"URL":"https:\/\/doi.org\/10.1007\/s00493-020-4657-0","relation":{},"ISSN":["0209-9683","1439-6912"],"issn-type":[{"value":"0209-9683","type":"print"},{"value":"1439-6912","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,3,14]]},"assertion":[{"value":"21 October 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 July 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 March 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}