{"id":"https://openalex.org/W2064317094","doi":"https://doi.org/10.1007/s00493-003-0028-x","title":"Sharp Concentration of the Number of Submaps in Random Planar Triangulations","display_name":"Sharp Concentration of the Number of Submaps in Random Planar Triangulations","publication_year":2003,"publication_date":"2003-07-01","ids":{"openalex":"https://openalex.org/W2064317094","doi":"https://doi.org/10.1007/s00493-003-0028-x","mag":"2064317094"},"language":"en","primary_location":{"id":"doi:10.1007/s00493-003-0028-x","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s00493-003-0028-x","pdf_url":null,"source":{"id":"https://openalex.org/S72366220","display_name":"COMBINATORICA","issn_l":"0209-9683","issn":["0209-9683","1439-6912"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"COMBINATORICA","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/A5102714122","display_name":"Zhicheng Gao","orcid":"https://orcid.org/0000-0001-6488-8721"},"institutions":[{"id":"https://openalex.org/I67031392","display_name":"Carleton University","ror":"https://ror.org/02qtvee93","country_code":"CA","type":"education","lineage":["https://openalex.org/I67031392"]}],"countries":["CA"],"is_corresponding":true,"raw_author_name":"Zhicheng Gao*","raw_affiliation_strings":["Department of Mathematics and\nStatistics, Carleton University, Ottawa, K1S 5B6, CANADA","Department of Mathematics and#N#Statistics, Carleton University, Ottawa, CANADA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and\nStatistics, Carleton University, Ottawa, K1S 5B6, CANADA","institution_ids":["https://openalex.org/I67031392"]},{"raw_affiliation_string":"Department of Mathematics and#N#Statistics, Carleton University, Ottawa, CANADA","institution_ids":["https://openalex.org/I67031392"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5113900323","display_name":"Nicholas Wormald","orcid":null},"institutions":[{"id":"https://openalex.org/I151746483","display_name":"University of Waterloo","ror":"https://ror.org/01aff2v68","country_code":"CA","type":"education","lineage":["https://openalex.org/I151746483"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Nicholas C. Wormald\u2020","raw_affiliation_strings":["Dept. of Combinatorics and\nOptimization, University of Waterloo, Waterloo, ON, N22 341, CANADA","Dept. of Combinatorics and#N#Optimization, University of Waterloo, Waterloo, CANADA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Dept. of Combinatorics and\nOptimization, University of Waterloo, Waterloo, ON, N22 341, CANADA","institution_ids":["https://openalex.org/I151746483"]},{"raw_affiliation_string":"Dept. of Combinatorics and#N#Optimization, University of Waterloo, Waterloo, CANADA","institution_ids":["https://openalex.org/I151746483"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5102714122"],"corresponding_institution_ids":["https://openalex.org/I67031392"],"apc_list":null,"apc_paid":null,"fwci":0.3359,"has_fulltext":false,"cited_by_count":20,"citation_normalized_percentile":{"value":0.70407524,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"23","issue":"3","first_page":"467","last_page":"486"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9987999796867371,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9987999796867371,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11106","display_name":"Data Management and Algorithms","score":0.9929999709129333,"subfield":{"id":"https://openalex.org/subfields/1711","display_name":"Signal Processing"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T14339","display_name":"Image Processing and 3D Reconstruction","score":0.9666000008583069,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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/mathematics","display_name":"Mathematics","score":0.8620511889457703},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.8345761299133301},{"id":"https://openalex.org/keywords/planar","display_name":"Planar","score":0.6731908321380615},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.6576521992683411},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.6129526495933533},{"id":"https://openalex.org/keywords/convex-polygon","display_name":"Convex polygon","score":0.5855212807655334},{"id":"https://openalex.org/keywords/polygon","display_name":"Polygon (computer graphics)","score":0.5732090473175049},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.5666764378547668},{"id":"https://openalex.org/keywords/limiting","display_name":"Limiting","score":0.48514923453330994},{"id":"https://openalex.org/keywords/triangulation","display_name":"Triangulation","score":0.47260552644729614},{"id":"https://openalex.org/keywords/planar-graph","display_name":"Planar graph","score":0.43627700209617615},{"id":"https://openalex.org/keywords/point-set-triangulation","display_name":"Point set triangulation","score":0.4341791868209839},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.2302008867263794},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.18657609820365906},{"id":"https://openalex.org/keywords/delaunay-triangulation","display_name":"Delaunay triangulation","score":0.12705421447753906},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.062009990215301514}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8620511889457703},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8345761299133301},{"id":"https://openalex.org/C134786449","wikidata":"https://www.wikidata.org/wiki/Q3391255","display_name":"Planar","level":2,"score":0.6731908321380615},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.6576521992683411},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.6129526495933533},{"id":"https://openalex.org/C2780034980","wikidata":"https://www.wikidata.org/wiki/Q3259505","display_name":"Convex polygon","level":3,"score":0.5855212807655334},{"id":"https://openalex.org/C190694206","wikidata":"https://www.wikidata.org/wiki/Q3276654","display_name":"Polygon (computer graphics)","level":3,"score":0.5732090473175049},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.5666764378547668},{"id":"https://openalex.org/C188198153","wikidata":"https://www.wikidata.org/wiki/Q1613840","display_name":"Limiting","level":2,"score":0.48514923453330994},{"id":"https://openalex.org/C135981907","wikidata":"https://www.wikidata.org/wiki/Q188056","display_name":"Triangulation","level":2,"score":0.47260552644729614},{"id":"https://openalex.org/C101837359","wikidata":"https://www.wikidata.org/wiki/Q547823","display_name":"Planar graph","level":3,"score":0.43627700209617615},{"id":"https://openalex.org/C167117609","wikidata":"https://www.wikidata.org/wiki/Q17078161","display_name":"Point set triangulation","level":3,"score":0.4341791868209839},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.2302008867263794},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.18657609820365906},{"id":"https://openalex.org/C68010082","wikidata":"https://www.wikidata.org/wiki/Q192445","display_name":"Delaunay triangulation","level":2,"score":0.12705421447753906},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.062009990215301514},{"id":"https://openalex.org/C126042441","wikidata":"https://www.wikidata.org/wiki/Q1324888","display_name":"Frame (networking)","level":2,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C76155785","wikidata":"https://www.wikidata.org/wiki/Q418","display_name":"Telecommunications","level":1,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","level":1,"score":0.0},{"id":"https://openalex.org/C121684516","wikidata":"https://www.wikidata.org/wiki/Q7600677","display_name":"Computer graphics (images)","level":1,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s00493-003-0028-x","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s00493-003-0028-x","pdf_url":null,"source":{"id":"https://openalex.org/S72366220","display_name":"COMBINATORICA","issn_l":"0209-9683","issn":["0209-9683","1439-6912"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"COMBINATORICA","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":13,"referenced_works":["https://openalex.org/W60767860","https://openalex.org/W1966197720","https://openalex.org/W2000336291","https://openalex.org/W2012068660","https://openalex.org/W2030997953","https://openalex.org/W2042987883","https://openalex.org/W2045390532","https://openalex.org/W2052888479","https://openalex.org/W2057116368","https://openalex.org/W2057534454","https://openalex.org/W2087428017","https://openalex.org/W2095339031","https://openalex.org/W2798655805"],"related_works":["https://openalex.org/W2374493932","https://openalex.org/W2952554563","https://openalex.org/W2369500278","https://openalex.org/W1985165973","https://openalex.org/W2110610374","https://openalex.org/W313723820","https://openalex.org/W2951263013","https://openalex.org/W4298281251","https://openalex.org/W2039867343","https://openalex.org/W2369201210"],"abstract_inverted_index":null,"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":1},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":1}],"updated_date":"2026-06-19T15:47:20.252518","created_date":"2025-10-10T00:00:00"}