{"id":"https://openalex.org/W2130083422","doi":"https://doi.org/10.1017/s0963548310000155","title":"A Threshold Phenomenon for Random Independent Sets in the Discrete Hypercube","display_name":"A Threshold Phenomenon for Random Independent Sets in the Discrete Hypercube","publication_year":2010,"publication_date":"2010-07-02","ids":{"openalex":"https://openalex.org/W2130083422","doi":"https://doi.org/10.1017/s0963548310000155","mag":"2130083422"},"language":"en","primary_location":{"id":"doi:10.1017/s0963548310000155","is_oa":true,"landing_page_url":"https://doi.org/10.1017/s0963548310000155","pdf_url":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/2267E1E6F3D7B64A6457B71980EF7967/S0963548310000155a.pdf/div-class-title-a-threshold-phenomenon-for-random-independent-sets-in-the-discrete-hypercube-div.pdf","source":{"id":"https://openalex.org/S7291079","display_name":"Combinatorics Probability Computing","issn_l":"0963-5483","issn":["0963-5483","1469-2163"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorics, Probability and Computing","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/2267E1E6F3D7B64A6457B71980EF7967/S0963548310000155a.pdf/div-class-title-a-threshold-phenomenon-for-random-independent-sets-in-the-discrete-hypercube-div.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5073164280","display_name":"David Galvin","orcid":"https://orcid.org/0000-0002-9777-9852"},"institutions":[{"id":"https://openalex.org/I107639228","display_name":"University of Notre Dame","ror":"https://ror.org/00mkhxb43","country_code":"US","type":"education","lineage":["https://openalex.org/I107639228"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"DAVID GALVIN","raw_affiliation_strings":["Department of Mathematics, University of Notre Dame, 255 Hurley Hall, Notre Dame, IN 46556, USA (e-mail:","Department of mathematics, university of notre dame, 255 hurley hall, notre dame, in 46556, usa (e-mail: dgalvin1@nd.edu)#TAB#"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Notre Dame, 255 Hurley Hall, Notre Dame, IN 46556, USA (e-mail:","institution_ids":["https://openalex.org/I107639228"]},{"raw_affiliation_string":"Department of mathematics, university of notre dame, 255 hurley hall, notre dame, in 46556, usa (e-mail: dgalvin1@nd.edu)#TAB#","institution_ids":["https://openalex.org/I107639228"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5073164280"],"corresponding_institution_ids":["https://openalex.org/I107639228"],"apc_list":null,"apc_paid":null,"fwci":1.6299,"has_fulltext":true,"cited_by_count":19,"citation_normalized_percentile":{"value":0.82066202,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"20","issue":"1","first_page":"27","last_page":"51"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.9991999864578247,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.9991999864578247,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":0.9988999962806702,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9975000023841858,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7893393039703369},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7447997331619263},{"id":"https://openalex.org/keywords/hypercube","display_name":"Hypercube","score":0.7091572284698486},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.5327969193458557},{"id":"https://openalex.org/keywords/distribution","display_name":"Distribution (mathematics)","score":0.5222221612930298},{"id":"https://openalex.org/keywords/partition","display_name":"Partition (number theory)","score":0.48107290267944336},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4414578378200531},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.4322054386138916},{"id":"https://openalex.org/keywords/matching","display_name":"Matching (statistics)","score":0.41905373334884644},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.15772724151611328},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.15014499425888062},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.08496156334877014}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7893393039703369},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7447997331619263},{"id":"https://openalex.org/C50820777","wikidata":"https://www.wikidata.org/wiki/Q213723","display_name":"Hypercube","level":2,"score":0.7091572284698486},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.5327969193458557},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.5222221612930298},{"id":"https://openalex.org/C42812","wikidata":"https://www.wikidata.org/wiki/Q1082910","display_name":"Partition (number theory)","level":2,"score":0.48107290267944336},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4414578378200531},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.4322054386138916},{"id":"https://openalex.org/C165064840","wikidata":"https://www.wikidata.org/wiki/Q1321061","display_name":"Matching (statistics)","level":2,"score":0.41905373334884644},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.15772724151611328},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.15014499425888062},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.08496156334877014}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1017/s0963548310000155","is_oa":true,"landing_page_url":"https://doi.org/10.1017/s0963548310000155","pdf_url":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/2267E1E6F3D7B64A6457B71980EF7967/S0963548310000155a.pdf/div-class-title-a-threshold-phenomenon-for-random-independent-sets-in-the-discrete-hypercube-div.pdf","source":{"id":"https://openalex.org/S7291079","display_name":"Combinatorics Probability Computing","issn_l":"0963-5483","issn":["0963-5483","1469-2163"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorics, Probability and Computing","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1017/s0963548310000155","is_oa":true,"landing_page_url":"https://doi.org/10.1017/s0963548310000155","pdf_url":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/2267E1E6F3D7B64A6457B71980EF7967/S0963548310000155a.pdf/div-class-title-a-threshold-phenomenon-for-random-independent-sets-in-the-discrete-hypercube-div.pdf","source":{"id":"https://openalex.org/S7291079","display_name":"Combinatorics Probability Computing","issn_l":"0963-5483","issn":["0963-5483","1469-2163"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorics, Probability and Computing","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2809240623","display_name":null,"funder_award_id":"H98230-10-1-0364","funder_id":"https://openalex.org/F4320311089","funder_display_name":"National Security Agency"},{"id":"https://openalex.org/G7845798442","display_name":null,"funder_award_id":"H98230","funder_id":"https://openalex.org/F4320311089","funder_display_name":"National Security Agency"}],"funders":[{"id":"https://openalex.org/F4320311089","display_name":"National Security Agency","ror":"https://ror.org/0047bvr32"},{"id":"https://openalex.org/F4320337380","display_name":"Division of Mathematical Sciences","ror":"https://ror.org/051fftw81"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2130083422.pdf","grobid_xml":"https://content.openalex.org/works/W2130083422.grobid-xml"},"referenced_works_count":13,"referenced_works":["https://openalex.org/W1986646471","https://openalex.org/W2001556264","https://openalex.org/W2004308928","https://openalex.org/W2052333181","https://openalex.org/W2063177160","https://openalex.org/W2078000356","https://openalex.org/W2100020751","https://openalex.org/W2103012681","https://openalex.org/W2122908915","https://openalex.org/W2163322487","https://openalex.org/W2952032359","https://openalex.org/W4233413206","https://openalex.org/W4237496336"],"related_works":["https://openalex.org/W2916710807","https://openalex.org/W4286907761","https://openalex.org/W2413588304","https://openalex.org/W3205447645","https://openalex.org/W2009411567","https://openalex.org/W2371629703","https://openalex.org/W2097769911","https://openalex.org/W2013252038","https://openalex.org/W2386598361","https://openalex.org/W2078375028"],"abstract_inverted_index":{"Let":[0],"I":[1,38,47,63,70,74,99,103,155,160,237,264,275],"be":[2],"an":[3,118,135,214],"independent":[4,36,147,215,242],"set":[5,37,154,216],"drawn":[6,217],"from":[7,240],"the":[8,21,31,60,86,132,144,163,169,211,220,241,249,267],"discrete":[9],"d":[10,14,18,91,112,126,142,151,176,246],"-dimensional":[11],"hypercube":[12],"Q":[13,90,141,150,175,245],"=":[15,57,77],"{0,":[16],"1}":[17],"according":[19,218,247],"to":[20,44,203,219,248],"hard-core":[22,164,221,250],"distribution":[23,32,251],"with":[24,41,152,252],"parameter":[25,253],"\u03bb":[26,45,56,66,95,139,158,173,233],">":[27,67,234],"0":[28,78],"(that":[29],"is,":[30],"in":[33,59,111,117,131,149,263,274],"which":[34],"each":[35,153],"is":[39,106,122,134,238,273,276],"chosen":[40,239],"probability":[42,268],"proportional":[43],"|":[46,48,73,102,159,161],").":[49],"We":[50,167,223],"show":[51],"a":[52,226,257],"sharp":[53],"transition":[54,115],"around":[55],"1":[58],"appearance":[61],"of":[62,89,123,137,171,195,213,244],":":[64],"for":[65,94,178,188,279,284],"1,":[68,97],"min{|":[69,98],"\u2229":[71,75,100,104],"\u0190|,":[72,101],"|}":[76,105],"asymptotically":[79,107],"almost":[80,108],"surely,":[81],"where":[82],"\u0190":[83,261],"and":[84,181,185,197],"are":[85],"bipartition":[87],"classes":[88],",":[92,180,192,265],"whereas":[93],"<":[96],"surely":[109],"exponential":[110],".":[113,127],"The":[114],"occurs":[116],"interval":[119],"whose":[120],"length":[121],"order":[124],"1/":[125],"A":[128],"key":[129],"step":[130],"proof":[133],"estimation":[136],"Z":[138,172],"(":[140,174],"),":[143],"sum":[145],"over":[146],"sets":[148,243],"given":[156],"weight":[157],"(a.k.a.":[162],"partition":[165],"function).":[166],"obtain":[168],"asymptotics":[170],")":[177],"$\\gl>\\sqrt{2}-1$":[179],"nearly":[182],"matching":[183],"upper":[184],"lower":[186],"bounds":[187,200],"$\\gl":[189],"\\leq":[190],"\\sqrt{2}-1$":[191],"extending":[193],"work":[194],"Korshunov":[196],"Sapozhenko.":[198],"These":[199],"allow":[201],"us":[202],"read":[204],"off":[205],"some":[206],"very":[207],"specific":[208],"information":[209],"about":[210],"structure":[212],"distribution.":[222],"also":[224],"derive":[225],"long-range":[227],"influence":[228],"result.":[229],"For":[230],"all":[231],"fixed":[232],"0,":[235],"if":[236],"\u03bb,":[254],"conditioned":[255],"on":[256],"particular":[258],"v":[259],"\u2208":[260,281,286],"being":[262],"then":[266],"that":[269],"another":[270],"vertex":[271],"w":[272,280,285],"o":[277],"(1)":[278],"but":[282],"\u03a9(1)":[283],"\u0190.":[287]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":1},{"year":2012,"cited_by_count":2}],"updated_date":"2026-03-15T09:29:46.208133","created_date":"2025-10-10T00:00:00"}