{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,1]],"date-time":"2022-04-01T14:02:25Z","timestamp":1648821745704},"reference-count":13,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2008,9,12]],"date-time":"2008-09-12T00:00:00Z","timestamp":1221177600000},"content-version":"unspecified","delay-in-days":5582,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[1993,6]]},"abstract":"<jats:p>For <jats:italic>T<\/jats:italic> \u2208 <jats:italic>GL<jats:sub>n<\/jats:sub><\/jats:italic> (<jats:italic>F<jats:sub>q<\/jats:sub><\/jats:italic>), let \u03a9<jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> (<jats:italic>t, T<\/jats:italic>) be the number of irreducible factors of degree less than or equal to <jats:italic>n<jats:sup>t<\/jats:sup><\/jats:italic> in the characteristic polynomial of <jats:italic>T<\/jats:italic>. Let<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0963548300000572eqnU1\" \/><\/jats:disp-formula><\/jats:p><jats:p>and suppose <jats:italic>T<\/jats:italic> is chosen from <jats:italic>G L<jats:sub>n<\/jats:sub><\/jats:italic>(<jats:italic>F<jats:sub>q<\/jats:sub><\/jats:italic>) at random uniformly. We prove that the stochastic process \u227a<jats:italic>Z<jats:sub>n<\/jats:sub><\/jats:italic>(<jats:italic>t<\/jats:italic>)\u227b<jats:sub><jats:italic>t<\/jats:italic>\u2208[0, 1]<\/jats:sub> converges to the standard Brownian motion process <jats:italic>W<\/jats:italic>(<jats:italic>t<\/jats:italic>), as n \u2192 \u221e.<\/jats:p>","DOI":"10.1017\/s0963548300000572","type":"journal-article","created":{"date-parts":[[2008,9,12]],"date-time":"2008-09-12T11:19:08Z","timestamp":1221218348000},"page":"157-180","source":"Crossref","is-referenced-by-count":4,"title":["Random Matrices and Brownian Motion"],"prefix":"10.1017","volume":"2","author":[{"given":"William M. Y.","family":"Goh","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Eric","family":"Schmutz","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2008,9,12]]},"reference":[{"key":"S0963548300000572_ref012","first-page":"242","volume-title":"A course of modern analysis","author":"Whitaker","year":"1961"},{"key":"S0963548300000572_ref010","first-page":"115","volume-title":"The theory of error correcting codes","author":"MacWilliams","year":"1981"},{"key":"S0963548300000572_ref008","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(81)90227-5"},{"key":"S0963548300000572_ref007","unstructured":"[7] Hansen J. C. (to appear) Combinatorics Probability and Computing."},{"key":"S0963548300000572_ref006","unstructured":"[6] Hansen J. C. (preprint) A functional central limit theorem for random mappings."},{"key":"S0963548300000572_ref004","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.3240020105"},{"key":"S0963548300000572_ref002","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1985.119.287"},{"key":"S0963548300000572_ref001","volume-title":"Convergence of probability Measures","author":"Billingsley","year":"1968"},{"key":"S0963548300000572_ref005","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/0377-0427(92)90251-R","article-title":"Affine walk on the hypercube","volume":"41","author":"Graham","year":"1992","journal-title":"Journal of Computational and Applied Mathematics"},{"key":"S0963548300000572_ref011","doi-asserted-by":"publisher","DOI":"10.1016\/0196-8858(88)90012-7"},{"key":"S0963548300000572_ref003","volume-title":"The theory of stochastic processes","author":"Gihman","year":"1974"},{"key":"S0963548300000572_ref013","volume-title":"Trigonometric series","author":"Zygmund","year":"1959"},{"key":"S0963548300000572_ref009","volume-title":"Encyclopedia of Mathematics and its Applications","volume":"20","author":"Lidl","year":"1983"}],"container-title":["Combinatorics, Probability and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0963548300000572","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,15]],"date-time":"2019-05-15T22:49:11Z","timestamp":1557960551000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0963548300000572\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1993,6]]},"references-count":13,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1993,6]]}},"alternative-id":["S0963548300000572"],"URL":"https:\/\/doi.org\/10.1017\/s0963548300000572","relation":{},"ISSN":["0963-5483","1469-2163"],"issn-type":[{"value":"0963-5483","type":"print"},{"value":"1469-2163","type":"electronic"}],"subject":[],"published":{"date-parts":[[1993,6]]}}}