{"year":"2025","has_accepted_license":"1","oa_version":"Published Version","corr_author":"1","publisher":"European Mathematical Society Press","PlanS_conform":"1","external_id":{"arxiv":["2303.05435"]},"DOAJ_listed":"1","department":[{"_id":"MaKw"}],"date_updated":"2026-02-23T10:44:41Z","date_created":"2026-02-17T07:41:59Z","publication_identifier":{"eissn":["1435-9863"],"issn":["1435-9855"]},"publication_status":"epub_ahead","article_type":"original","author":[{"first_name":"Margalit","last_name":"Glasgow","full_name":"Glasgow, Margalit"},{"full_name":"Kwan, Matthew Alan","first_name":"Matthew Alan","last_name":"Kwan","orcid":"0000-0002-4003-7567","id":"5fca0887-a1db-11eb-95d1-ca9d5e0453b3"},{"last_name":"Sah","first_name":"Ashwin","full_name":"Sah, Ashwin"},{"full_name":"Sawhney, Mehtaab","first_name":"Mehtaab","last_name":"Sawhney"}],"day":"02","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_processing_charge":"No","publication":"Journal of the European Mathematical Society","type":"journal_article","doi":"10.4171/jems/1692","month":"09","title":"The exact rank of sparse random graphs","language":[{"iso":"eng"}],"oa":1,"acknowledgement":"We would like to thank Noga Alon for suggesting that our main result gives\r\na linear-time algorithm for computing the rank. We also thank the referees for a number of thoughtful comments and suggestions. Glasgow was supported by NSF graduate research fellowship program award DGE1656518. Kwan was supported by ERC Starting Grant “RANDSTRUCT” No. 101076777. Sah and Sawhney were supported by NSF Graduate Research Fellowship Program DGE-1745302.\r\n","status":"public","date_published":"2025-09-02T00:00:00Z","quality_controlled":"1","OA_type":"diamond","abstract":[{"text":"Two landmark results in combinatorial random matrix theory, due to Komlós and Costello–Tao–Vu, show that discrete random matrices and symmetric discrete random matrices are typically nonsingular. In particular, in the language of graph theory, when p is a fixed constant, the biadjacency matrix of a random Erdős–Rényi bipartite graph G(n,n,p) and the adjacency matrix of an Erdős–Rényi random graph G(n,p) are both nonsingular with high probability. However, very sparse random graphs (i.e., where p is allowed to decay rapidly with n) are typically singular, due to the presence of “local” dependencies such as isolated vertices and pairs of degree-1 vertices with the same neighbour. In this paper, we give a combinatorial description of the rank of a sparse random graph G(n,n,c/n) or G(n,c/n) in terms of such local dependencies, for all constants c=e (and we present some evidence that the situation is very different for c=e). This gives an essentially complete answer to a question raised by Vu (2014). As applications of our main theorem and its proof, we also determine the asymptotic singularity probability of the 2-core of a sparse random graph, we show that the rank of a sparse random graph is extremely well approximated by its matching number, and we deduce a central limit theorem for the rank of G(n,c/n).","lang":"eng"}],"OA_place":"publisher","citation":{"short":"M. Glasgow, M.A. Kwan, A. Sah, M. Sawhney, Journal of the European Mathematical Society (2025).","chicago":"Glasgow, Margalit, Matthew Alan Kwan, Ashwin Sah, and Mehtaab Sawhney. “The Exact Rank of Sparse Random Graphs.” <i>Journal of the European Mathematical Society</i>. European Mathematical Society Press, 2025. <a href=\"https://doi.org/10.4171/jems/1692\">https://doi.org/10.4171/jems/1692</a>.","ama":"Glasgow M, Kwan MA, Sah A, Sawhney M. The exact rank of sparse random graphs. <i>Journal of the European Mathematical Society</i>. 2025. doi:<a href=\"https://doi.org/10.4171/jems/1692\">10.4171/jems/1692</a>","mla":"Glasgow, Margalit, et al. “The Exact Rank of Sparse Random Graphs.” <i>Journal of the European Mathematical Society</i>, European Mathematical Society Press, 2025, doi:<a href=\"https://doi.org/10.4171/jems/1692\">10.4171/jems/1692</a>.","ista":"Glasgow M, Kwan MA, Sah A, Sawhney M. 2025. The exact rank of sparse random graphs. Journal of the European Mathematical Society.","ieee":"M. Glasgow, M. A. Kwan, A. Sah, and M. Sawhney, “The exact rank of sparse random graphs,” <i>Journal of the European Mathematical Society</i>. European Mathematical Society Press, 2025.","apa":"Glasgow, M., Kwan, M. A., Sah, A., &#38; Sawhney, M. (2025). The exact rank of sparse random graphs. <i>Journal of the European Mathematical Society</i>. European Mathematical Society Press. <a href=\"https://doi.org/10.4171/jems/1692\">https://doi.org/10.4171/jems/1692</a>"},"project":[{"grant_number":"101076777","name":"Randomness and structure in combinatorics","_id":"bd95085b-d553-11ed-ba76-e55d3349be45"}],"main_file_link":[{"url":"https://doi.org/10.4171/JEMS/1692","open_access":"1"}],"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"ddc":["510"],"arxiv":1,"_id":"21263"}