{"id":"https://openalex.org/W3000106173","doi":"https://doi.org/10.37236/8092","title":"On the Rank of a Random Binary Matrix","display_name":"On the Rank of a Random Binary Matrix","publication_year":2019,"publication_date":"2019-10-11","ids":{"openalex":"https://openalex.org/W3000106173","doi":"https://doi.org/10.37236/8092","mag":"3000106173"},"language":"en","primary_location":{"id":"doi:10.37236/8092","is_oa":true,"landing_page_url":"https://doi.org/10.37236/8092","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v26i4p12/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v26i4p12/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5022238950","display_name":"Colin Cooper","orcid":"https://orcid.org/0000-0002-5264-4401"},"institutions":[{"id":"https://openalex.org/I164213776","display_name":"King's College - North Carolina","ror":"https://ror.org/022f44628","country_code":"US","type":"education","lineage":["https://openalex.org/I164213776"]},{"id":"https://openalex.org/I183935753","display_name":"King's College London","ror":"https://ror.org/0220mzb33","country_code":"GB","type":"education","lineage":["https://openalex.org/I124357947","https://openalex.org/I183935753"]}],"countries":["GB","US"],"is_corresponding":true,"raw_author_name":"Colin Cooper","raw_affiliation_strings":["Kings College, London"],"affiliations":[{"raw_affiliation_string":"Kings College, London","institution_ids":["https://openalex.org/I164213776","https://openalex.org/I183935753"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5056950019","display_name":"Alan Frieze","orcid":"https://orcid.org/0000-0002-8481-5615"},"institutions":[{"id":"https://openalex.org/I74973139","display_name":"Carnegie Mellon University","ror":"https://ror.org/05x2bcf33","country_code":"US","type":"education","lineage":["https://openalex.org/I74973139"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Alan Frieze","raw_affiliation_strings":["Carnegie Mellon University"],"affiliations":[{"raw_affiliation_string":"Carnegie Mellon University","institution_ids":["https://openalex.org/I74973139"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5054318475","display_name":"Wesley Pegden","orcid":"https://orcid.org/0000-0002-3680-1817"},"institutions":[{"id":"https://openalex.org/I74973139","display_name":"Carnegie Mellon University","ror":"https://ror.org/05x2bcf33","country_code":"US","type":"education","lineage":["https://openalex.org/I74973139"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Wesley Pegden","raw_affiliation_strings":["Carnegie Mellon University"],"affiliations":[{"raw_affiliation_string":"Carnegie Mellon University","institution_ids":["https://openalex.org/I74973139"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5022238950"],"corresponding_institution_ids":["https://openalex.org/I164213776","https://openalex.org/I183935753"],"apc_list":null,"apc_paid":null,"fwci":0.3801,"has_fulltext":true,"cited_by_count":6,"citation_normalized_percentile":{"value":0.62328529,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":96},"biblio":{"volume":"26","issue":"4","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11476","display_name":"Graph theory and applications","score":0.9890999794006348,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11476","display_name":"Graph theory and applications","score":0.9890999794006348,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11716","display_name":"Random Matrices and Applications","score":0.987500011920929,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12536","display_name":"Topological and Geometric Data Analysis","score":0.9779999852180481,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.878288984298706},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8095505237579346},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.6851011514663696},{"id":"https://openalex.org/keywords/hypergraph","display_name":"Hypergraph","score":0.548892617225647},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.5176073312759399},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.45648691058158875},{"id":"https://openalex.org/keywords/random-matrix","display_name":"Random matrix","score":0.44306662678718567},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3438725769519806},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.12960785627365112},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.10812333226203918},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.08651509881019592}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.878288984298706},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8095505237579346},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.6851011514663696},{"id":"https://openalex.org/C2781221856","wikidata":"https://www.wikidata.org/wiki/Q840247","display_name":"Hypergraph","level":2,"score":0.548892617225647},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5176073312759399},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.45648691058158875},{"id":"https://openalex.org/C64812099","wikidata":"https://www.wikidata.org/wiki/Q176604","display_name":"Random matrix","level":3,"score":0.44306662678718567},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3438725769519806},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.12960785627365112},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.10812333226203918},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.08651509881019592},{"id":"https://openalex.org/C159985019","wikidata":"https://www.wikidata.org/wiki/Q181790","display_name":"Composite material","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.37236/8092","is_oa":true,"landing_page_url":"https://doi.org/10.37236/8092","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v26i4p12/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.37236/8092","is_oa":true,"landing_page_url":"https://doi.org/10.37236/8092","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v26i4p12/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1159596240","display_name":null,"funder_award_id":"EP/M005038/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G2533871239","display_name":null,"funder_award_id":"DMS1363136","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G3229677935","display_name":"Randomized Algorithms for Computer Networks","funder_award_id":"EP/M005038/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G3330119427","display_name":null,"funder_award_id":"EPSRC","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G445480892","display_name":null,"funder_award_id":"DMS1661063","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G4797868072","display_name":"New directions arising from a special diffusion process on the integer lattice","funder_award_id":"1363136","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8139251166","display_name":"Random Structures and Algorithms","funder_award_id":"1661063","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3000106173.pdf","grobid_xml":"https://content.openalex.org/works/W3000106173.grobid-xml"},"referenced_works_count":10,"referenced_works":["https://openalex.org/W1885977048","https://openalex.org/W2001333579","https://openalex.org/W2003914969","https://openalex.org/W2054812794","https://openalex.org/W2223661604","https://openalex.org/W2962945180","https://openalex.org/W2966630122","https://openalex.org/W4238472918","https://openalex.org/W4253819323","https://openalex.org/W4299438412"],"related_works":["https://openalex.org/W4287692100","https://openalex.org/W3188963412","https://openalex.org/W2297336610","https://openalex.org/W4296581269","https://openalex.org/W4372262496","https://openalex.org/W3022715693","https://openalex.org/W1979883786","https://openalex.org/W2084962005","https://openalex.org/W2104364550","https://openalex.org/W2029664913"],"abstract_inverted_index":{"We":[0,48,189],"study":[1],"the":[2,24,39,55,61,77,99,104,110,117,129,142,159,163,196,202,208,224],"rank":[3,56,88,130,143],"of":[4,41,60,63,68,81,89,101,103,106,112,119,131,165,168,182,205,210,227],"a":[5,58,82,166,177,180,211,228],"random":[6,37,84,212],"$n":[7],"\\times":[8],"m$":[9],"matrix":[10,74,80],"$\\mathbf{A}_{n,m;k}$":[11,75,90,132],"with":[12],"entries":[13,20,26],"from":[14,38],"$GF(2)$,":[15],"and":[16,34,70,108,137,161],"exactly":[17],"$k$":[18],"unit":[19],"in":[21,66],"each":[22],"column,":[23],"other":[25],"being":[27],"zero.":[28],"The":[29,73,87],"columns":[30,64,169],"are":[31],"chosen":[32],"independently":[33],"uniformly":[35],"at":[36],"set":[40,167,181],"all":[42],"${n":[43],"\\choose":[44],"k}$":[45],"such":[46],"columns.":[47,188],"obtain":[49,190],"an":[50,191],"asymptotically":[51,192],"correct":[52,193],"estimate":[53,194],"for":[54,121,127,138,195,222],"as":[57,94,171,179],"function":[59],"number":[62,100,111],"$m$":[65,120],"terms":[67],"$c,n,k$,":[69],"where":[71],"$m=cn/k$.":[72],"forms":[76],"vertex-edge":[78],"incidence":[79],"$k$-uniform":[83],"hypergraph":[85],"$H$.":[86],"can":[91],"be":[92,98,116],"expressed":[93],"follows.":[95],"Let":[96,114],"$|C_2|$":[97],"vertices":[102],"2-core":[105],"$H$,":[107],"$|E(C_2)|$":[109],"edges.":[113],"$m^*$":[115],"value":[118,209],"which":[122],"$|C_2|=":[123],"|E(C_2)|$.":[124],"Then":[125],"w.h.p.":[126],"$m<m^*$":[128],"is":[133,144],"asymptotic":[134,145],"to":[135,146,158,234],"$m$,":[136],"$m":[139],"\\ge":[140],"m^*$":[141],"$m-|E(C_2)|+|C_2|$.":[147],"In":[148],"addition,":[149],"assign":[150],"i.i.d.":[151],"$U[0,1]$":[152],"weights":[153],"$X_i,":[154],"i":[155],"\\in":[156,173],"{1,2,...m}$":[157],"columns,":[160],"define":[162],"weight":[164,198,230],"$S$":[170],"$X(S)=\\sum_{j":[172],"S}":[174],"X_j$.":[175],"Define":[176],"basis":[178],"$n-\ud835\udfd9":[183],"(k\\text{":[184],"even})$":[185],"linearly":[186],"independent":[187],"minimum":[197,213,229],"basis.":[199],"This":[200],"generalises":[201],"well-known":[203],"result":[204],"Frieze":[206],"[On":[207],"spanning":[214,231],"tree":[215,232],"problem,":[216],"Discrete":[217],"Applied":[218],"Mathematics,":[219],"(1985)]":[220],"that,":[221],"$k=2$,":[223],"expected":[225],"length":[226],"tends":[233],"$\\zeta(3)\\sim":[235],"1.202$.":[236]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":1}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}