{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,25]],"date-time":"2025-02-25T05:25:06Z","timestamp":1740461106320,"version":"3.37.3"},"reference-count":33,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2024,10,18]],"date-time":"2024-10-18T00:00:00Z","timestamp":1729209600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2025,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The <jats:italic>walk matrix<\/jats:italic> associated to an <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000312_inline1.png\"\/><jats:tex-math>\n$n\\times n$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> integer matrix <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000312_inline2.png\"\/><jats:tex-math>\n$\\mathbf{X}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> and an integer vector <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000312_inline3.png\"\/><jats:tex-math>\n$b$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is defined by <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000312_inline4.png\"\/><jats:tex-math>\n${\\mathbf{W}} \\,:\\!=\\, (b,{\\mathbf{X}} b,\\ldots, {\\mathbf{X}}^{n-1}b)$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. We study limiting laws for the cokernel of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000312_inline5.png\"\/><jats:tex-math>\n$\\mathbf{W}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> in the scenario where <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000312_inline6.png\"\/><jats:tex-math>\n$\\mathbf{X}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is a random matrix with independent entries and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000312_inline7.png\"\/><jats:tex-math>\n$b$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is deterministic. Our first main result provides a formula for the distribution of the <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000312_inline8.png\"\/><jats:tex-math>\n$p^m$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-torsion part of the cokernel, as a group, when <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000312_inline9.png\"\/><jats:tex-math>\n$\\mathbf{X}$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> has independent entries from a specific distribution. The second main result relaxes the distributional assumption and concerns the <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0963548324000312_inline10.png\"\/><jats:tex-math>\n${\\mathbb{Z}}[x]$\n<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-module structure.<\/jats:p><jats:p>The motivation for this work arises from an open problem in spectral graph theory, which asks to show that random graphs are often determined up to isomorphism by their (generalised) spectrum. Sufficient conditions for generalised spectral determinacy can, namely, be stated in terms of the cokernel of a walk matrix. Extensions of our results could potentially be used to determine how often those conditions are satisfied. Some remaining challenges for such extensions are outlined in the paper.<\/jats:p>","DOI":"10.1017\/s0963548324000312","type":"journal-article","created":{"date-parts":[[2024,10,18]],"date-time":"2024-10-18T08:34:58Z","timestamp":1729240498000},"page":"131-150","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["Cokernel statistics for walk matrices of directed and weighted random graphs"],"prefix":"10.1017","volume":"34","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4670-1836","authenticated-orcid":false,"given":"Alexander","family":"Van Werde","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2024,10,18]]},"reference":[{"doi-asserted-by":"publisher","key":"S0963548324000312_ref11","DOI":"10.1093\/qmath\/haae030"},{"doi-asserted-by":"publisher","key":"S0963548324000312_ref17","DOI":"10.1016\/j.laa.2021.03.033"},{"doi-asserted-by":"publisher","key":"S0963548324000312_ref18","DOI":"10.1016\/j.disc.2022.113177"},{"volume-title":"Number Theory Noordwijkerhout 1983","year":"1984","author":"Cohen","key":"S0963548324000312_ref5"},{"doi-asserted-by":"publisher","key":"S0963548324000312_ref27","DOI":"10.1016\/j.jctb.2016.07.004"},{"unstructured":"[20] Sawin, W. and Wood, M. 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