{"id":"https://openalex.org/W2068593432","doi":"https://doi.org/10.1017/s0963548302005345","title":"An Improved Upper Bound for the Hexagonal Lattice Site Percolation Critical Probability","display_name":"An Improved Upper Bound for the Hexagonal Lattice Site Percolation Critical Probability","publication_year":2002,"publication_date":"2002-11-01","ids":{"openalex":"https://openalex.org/W2068593432","doi":"https://doi.org/10.1017/s0963548302005345","mag":"2068593432"},"language":"en","primary_location":{"id":"doi:10.1017/s0963548302005345","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0963548302005345","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/A5064543263","display_name":"John C. Wierman","orcid":"https://orcid.org/0000-0002-5000-9455"},"institutions":[{"id":"https://openalex.org/I145311948","display_name":"Johns Hopkins University","ror":"https://ror.org/00za53h95","country_code":"US","type":"education","lineage":["https://openalex.org/I145311948"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"JOHN C. WIERMAN","raw_affiliation_strings":["Mathematical Sciences Department, 104 Whitehead Hall, Johns Hopkins University, Baltimore, Maryland, MD 21218-2682, USA (e-mail:","Mathematical Sciences Department, 104 Whitehead Hall, Johns Hopkins University, Baltimore, Maryland, MD 21218-2682, USA (e-mail: wierman@jhu.edu)#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Mathematical Sciences Department, 104 Whitehead Hall, Johns Hopkins University, Baltimore, Maryland, MD 21218-2682, USA (e-mail:","institution_ids":["https://openalex.org/I145311948"]},{"raw_affiliation_string":"Mathematical Sciences Department, 104 Whitehead Hall, Johns Hopkins University, Baltimore, Maryland, MD 21218-2682, USA (e-mail: wierman@jhu.edu)#TAB#","institution_ids":["https://openalex.org/I145311948"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5064543263"],"corresponding_institution_ids":["https://openalex.org/I145311948"],"apc_list":null,"apc_paid":null,"fwci":4.163,"has_fulltext":false,"cited_by_count":21,"citation_normalized_percentile":{"value":0.94653722,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"11","issue":"6","first_page":"629","last_page":"643"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.9987000226974487,"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":0.9987000226974487,"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/T11476","display_name":"Graph theory and applications","score":0.9954000115394592,"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/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":0.9934999942779541,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.7587980031967163},{"id":"https://openalex.org/keywords/hexagonal-lattice","display_name":"Hexagonal lattice","score":0.690711498260498},{"id":"https://openalex.org/keywords/computation","display_name":"Computation","score":0.664673388004303},{"id":"https://openalex.org/keywords/lattice","display_name":"Lattice (music)","score":0.6258792281150818},{"id":"https://openalex.org/keywords/hexagonal-crystal-system","display_name":"Hexagonal crystal system","score":0.5944022536277771},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.5198317170143127},{"id":"https://openalex.org/keywords/percolation","display_name":"Percolation (cognitive psychology)","score":0.5186735391616821},{"id":"https://openalex.org/keywords/percolation-threshold","display_name":"Percolation threshold","score":0.45898330211639404},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.43841487169265747},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.43038347363471985},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.30562061071395874},{"id":"https://openalex.org/keywords/condensed-matter-physics","display_name":"Condensed matter physics","score":0.2578013837337494},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.16700494289398193},{"id":"https://openalex.org/keywords/crystallography","display_name":"Crystallography","score":0.16208761930465698},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.13248342275619507},{"id":"https://openalex.org/keywords/chemistry","display_name":"Chemistry","score":0.12043854594230652},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.10343989729881287},{"id":"https://openalex.org/keywords/electrical-resistivity-and-conductivity","display_name":"Electrical resistivity and conductivity","score":0.0791986882686615}],"concepts":[{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.7587980031967163},{"id":"https://openalex.org/C90626534","wikidata":"https://www.wikidata.org/wiki/Q4523381","display_name":"Hexagonal lattice","level":3,"score":0.690711498260498},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.664673388004303},{"id":"https://openalex.org/C2781204021","wikidata":"https://www.wikidata.org/wiki/Q6497091","display_name":"Lattice (music)","level":2,"score":0.6258792281150818},{"id":"https://openalex.org/C128765274","wikidata":"https://www.wikidata.org/wiki/Q663314","display_name":"Hexagonal crystal system","level":2,"score":0.5944022536277771},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.5198317170143127},{"id":"https://openalex.org/C2780457167","wikidata":"https://www.wikidata.org/wiki/Q17156484","display_name":"Percolation (cognitive psychology)","level":2,"score":0.5186735391616821},{"id":"https://openalex.org/C2729557","wikidata":"https://www.wikidata.org/wiki/Q7167170","display_name":"Percolation threshold","level":3,"score":0.45898330211639404},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.43841487169265747},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.43038347363471985},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.30562061071395874},{"id":"https://openalex.org/C26873012","wikidata":"https://www.wikidata.org/wiki/Q214781","display_name":"Condensed matter physics","level":1,"score":0.2578013837337494},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.16700494289398193},{"id":"https://openalex.org/C8010536","wikidata":"https://www.wikidata.org/wiki/Q160398","display_name":"Crystallography","level":1,"score":0.16208761930465698},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.13248342275619507},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.12043854594230652},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.10343989729881287},{"id":"https://openalex.org/C69990965","wikidata":"https://www.wikidata.org/wiki/Q65402698","display_name":"Electrical resistivity and conductivity","level":2,"score":0.0791986882686615},{"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},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"id":"https://openalex.org/C155355069","wikidata":"https://www.wikidata.org/wiki/Q575224","display_name":"Antiferromagnetism","level":2,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1017/s0963548302005345","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0963548302005345","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":[{"id":"https://openalex.org/F4320310145","display_name":"Johns Hopkins University","ror":"https://ror.org/00za53h95"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":13,"referenced_works":["https://openalex.org/W1594827281","https://openalex.org/W1610677200","https://openalex.org/W1972850107","https://openalex.org/W1996237450","https://openalex.org/W2039223991","https://openalex.org/W2049951355","https://openalex.org/W2054809932","https://openalex.org/W2058283287","https://openalex.org/W2066652472","https://openalex.org/W2074985975","https://openalex.org/W2116824607","https://openalex.org/W2143016004","https://openalex.org/W2146868839"],"related_works":["https://openalex.org/W2086354835","https://openalex.org/W2135100458","https://openalex.org/W2078697937","https://openalex.org/W1985161823","https://openalex.org/W2052206812","https://openalex.org/W2012801313","https://openalex.org/W2157530871","https://openalex.org/W1617582270","https://openalex.org/W2021404574","https://openalex.org/W2146411576"],"abstract_inverted_index":{"The":[0,22],"hexagonal":[1],"lattice":[2],"site":[3,34],"percolation":[4],"critical":[5],"probability":[6],"is":[7,24,44],"shown":[8],"to":[9,31],"be":[10],"at":[11],"most":[12],"0.79472,":[13],"improving":[14],"the":[15,28,33,37,40,56,59],"best":[16],"previous":[17],"mathematically":[18],"rigorous":[19],"upper":[20],"bound.":[21],"bound":[23,60],"derived":[25],"by":[26],"using":[27],"substitution":[29],"method":[30],"compare":[32],"model":[35],"with":[36],"bond":[38],"model,":[39],"latter":[41],"of":[42,53,58],"which":[43,48],"exactly":[45],"solved.":[46],"Shortcuts":[47],"eliminate":[49],"a":[50],"substantial":[51],"amount":[52],"computation":[54],"make":[55],"derivation":[57],"possible.":[61]},"counts_by_year":[{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}