{"id":"https://openalex.org/W2126552218","doi":"https://doi.org/10.1017/s0963548305007406","title":"Percolation on Finite Cayley Graphs","display_name":"Percolation on Finite Cayley Graphs","publication_year":2006,"publication_date":"2006-06-07","ids":{"openalex":"https://openalex.org/W2126552218","doi":"https://doi.org/10.1017/s0963548305007406","mag":"2126552218"},"language":"en","primary_location":{"id":"doi:10.1017/s0963548305007406","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0963548305007406","pdf_url":null,"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":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5075398549","display_name":"Christopher Malon","orcid":null},"institutions":[{"id":"https://openalex.org/I4210110987","display_name":"IIT@MIT","ror":"https://ror.org/01wp8zh54","country_code":"US","type":"facility","lineage":["https://openalex.org/I30771326","https://openalex.org/I4210110987"]},{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"CHRISTOPHER MALON","raw_affiliation_strings":["MIT Department of Mathematics, Cambridge, MA 02139, USA (e-mail:","MIT Department of Mathematics, Cambridge, MA 02139, USA (e-mail: malon@math.mit.edu, pak@math.mit.edu)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"MIT Department of Mathematics, Cambridge, MA 02139, USA (e-mail:","institution_ids":["https://openalex.org/I4210110987"]},{"raw_affiliation_string":"MIT Department of Mathematics, Cambridge, MA 02139, USA (e-mail: malon@math.mit.edu, pak@math.mit.edu)","institution_ids":["https://openalex.org/I63966007"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5080560017","display_name":"Igor Pak","orcid":"https://orcid.org/0000-0001-8579-7239"},"institutions":[{"id":"https://openalex.org/I4210110987","display_name":"IIT@MIT","ror":"https://ror.org/01wp8zh54","country_code":"US","type":"facility","lineage":["https://openalex.org/I30771326","https://openalex.org/I4210110987"]},{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"IGOR PAK","raw_affiliation_strings":["MIT Department of Mathematics, Cambridge, MA 02139, USA (e-mail:","MIT Department of Mathematics, Cambridge, MA 02139, USA (e-mail: malon@math.mit.edu, pak@math.mit.edu)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"MIT Department of Mathematics, Cambridge, MA 02139, USA (e-mail:","institution_ids":["https://openalex.org/I4210110987"]},{"raw_affiliation_string":"MIT Department of Mathematics, Cambridge, MA 02139, USA (e-mail: malon@math.mit.edu, pak@math.mit.edu)","institution_ids":["https://openalex.org/I63966007"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.3853,"has_fulltext":false,"cited_by_count":4,"citation_normalized_percentile":{"value":0.59979977,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"15","issue":"4","first_page":"571","last_page":"588"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.9988999962806702,"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":0.9988999962806702,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9986000061035156,"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"}},{"id":"https://openalex.org/T11476","display_name":"Graph theory and applications","score":0.9980999827384949,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/cayley-graph","display_name":"Cayley graph","score":0.960862398147583},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.6772618293762207},{"id":"https://openalex.org/keywords/percolation","display_name":"Percolation (cognitive psychology)","score":0.6583951711654663},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6170158386230469},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.6046918630599976},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5705488920211792},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.5352047085762024},{"id":"https://openalex.org/keywords/vertex-transitive-graph","display_name":"Vertex-transitive graph","score":0.5305070281028748},{"id":"https://openalex.org/keywords/transitive-relation","display_name":"Transitive relation","score":0.49629122018814087},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.45203083753585815},{"id":"https://openalex.org/keywords/cayley-transform","display_name":"Cayley transform","score":0.4512943923473358},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.43393999338150024},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.060465335845947266},{"id":"https://openalex.org/keywords/voltage-graph","display_name":"Voltage graph","score":0.04718366265296936}],"concepts":[{"id":"https://openalex.org/C120204988","wikidata":"https://www.wikidata.org/wiki/Q859174","display_name":"Cayley graph","level":3,"score":0.960862398147583},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.6772618293762207},{"id":"https://openalex.org/C2780457167","wikidata":"https://www.wikidata.org/wiki/Q17156484","display_name":"Percolation (cognitive psychology)","level":2,"score":0.6583951711654663},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6170158386230469},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.6046918630599976},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5705488920211792},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.5352047085762024},{"id":"https://openalex.org/C113203676","wikidata":"https://www.wikidata.org/wiki/Q220997","display_name":"Vertex-transitive graph","level":5,"score":0.5305070281028748},{"id":"https://openalex.org/C191399111","wikidata":"https://www.wikidata.org/wiki/Q64861","display_name":"Transitive relation","level":2,"score":0.49629122018814087},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.45203083753585815},{"id":"https://openalex.org/C18586221","wikidata":"https://www.wikidata.org/wiki/Q2778065","display_name":"Cayley transform","level":5,"score":0.4512943923473358},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.43393999338150024},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.060465335845947266},{"id":"https://openalex.org/C22149727","wikidata":"https://www.wikidata.org/wiki/Q7940747","display_name":"Voltage graph","level":4,"score":0.04718366265296936},{"id":"https://openalex.org/C203776342","wikidata":"https://www.wikidata.org/wiki/Q1378376","display_name":"Line graph","level":3,"score":0.0},{"id":"https://openalex.org/C169760540","wikidata":"https://www.wikidata.org/wiki/Q207011","display_name":"Neuroscience","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1017/s0963548305007406","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0963548305007406","pdf_url":null,"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":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":12,"referenced_works":["https://openalex.org/W1508349527","https://openalex.org/W1518422876","https://openalex.org/W1528152028","https://openalex.org/W2026620883","https://openalex.org/W2103528061","https://openalex.org/W2106699235","https://openalex.org/W2141899664","https://openalex.org/W2161792498","https://openalex.org/W2164314258","https://openalex.org/W2503931018","https://openalex.org/W3098555975","https://openalex.org/W3144881883"],"related_works":["https://openalex.org/W1968298352","https://openalex.org/W1985023158","https://openalex.org/W2601839913","https://openalex.org/W4213105785","https://openalex.org/W3024284632","https://openalex.org/W4294586788","https://openalex.org/W1777634055","https://openalex.org/W2348558285","https://openalex.org/W4298173932","https://openalex.org/W4226275486"],"abstract_inverted_index":{"In":[0],"this":[1],"paper,":[2],"we":[3],"study":[4],"percolation":[5,18],"on":[6],"finite":[7],"Cayley":[8,44,57],"graphs.":[9,45],"A":[10],"conjecture":[11,38],"of":[12,20,43],"Benjamini":[13],"says":[14],"that":[15],"the":[16],"critical":[17],"$p_c$":[19,64],"any":[21],"vertex-transitive":[22],"graph":[23],"satisfying":[24],"a":[25,49],"certain":[26],"diameter":[27],"condition":[28],"can":[29],"be":[30],"bounded":[31],"away":[32],"from":[33],"one.":[34],"We":[35,46],"prove":[36],"Benjamini's":[37],"for":[39,59],"some":[40],"special":[41],"classes":[42],"also":[47],"establish":[48],"reduction":[50],"theorem,":[51],"which":[52],"allows":[53],"us":[54],"to":[55],"build":[56],"graphs":[58],"large":[60],"groups":[61],"without":[62],"increasing":[63],".":[65]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2026-07-30T17:31:21.811387","created_date":"2025-10-10T00:00:00"}
