{"id":"https://openalex.org/W2736869835","doi":"https://doi.org/10.1007/s00454-019-00057-2","title":"Harmonic Dirichlet Functions on Planar Graphs","display_name":"Harmonic Dirichlet Functions on Planar Graphs","publication_year":2019,"publication_date":"2019-01-30","ids":{"openalex":"https://openalex.org/W2736869835","doi":"https://doi.org/10.1007/s00454-019-00057-2","mag":"2736869835"},"language":"en","primary_location":{"id":"doi:10.1007/s00454-019-00057-2","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00454-019-00057-2","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00454-019-00057-2.pdf","source":{"id":"https://openalex.org/S103671469","display_name":"Discrete & Computational Geometry","issn_l":"0179-5376","issn":["0179-5376","1432-0444"],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Discrete & Computational Geometry","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s00454-019-00057-2.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5060156386","display_name":"Tom Hutchcroft","orcid":"https://orcid.org/0000-0003-0061-593X"},"institutions":[{"id":"https://openalex.org/I241749","display_name":"University of Cambridge","ror":"https://ror.org/013meh722","country_code":"GB","type":"education","lineage":["https://openalex.org/I241749"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Tom Hutchcroft","raw_affiliation_strings":["Statslab, DPMMS, University of Cambridge, Cambridge, UK","Univ. of Cambridge"],"raw_orcid":"https://orcid.org/0000-0003-0061-593X","affiliations":[{"raw_affiliation_string":"Statslab, DPMMS, University of Cambridge, Cambridge, UK","institution_ids":["https://openalex.org/I241749"]},{"raw_affiliation_string":"Univ. of Cambridge","institution_ids":["https://openalex.org/I241749"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5060156386"],"corresponding_institution_ids":["https://openalex.org/I241749"],"apc_list":{"value":2290,"currency":"EUR","value_usd":2890},"apc_paid":{"value":2290,"currency":"EUR","value_usd":2890},"fwci":0.3787,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.4764104,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"61","issue":"3","first_page":"479","last_page":"506"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9980999827384949,"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/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9980999827384949,"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/T10304","display_name":"Geometric and Algebraic Topology","score":0.9979000091552734,"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":0.9976000189781189,"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/mathematics","display_name":"Mathematics","score":0.8214662075042725},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.7220975756645203},{"id":"https://openalex.org/keywords/harmonic-function","display_name":"Harmonic function","score":0.6925757527351379},{"id":"https://openalex.org/keywords/dirichlet-problem","display_name":"Dirichlet problem","score":0.45065802335739136},{"id":"https://openalex.org/keywords/harmonic","display_name":"Harmonic","score":0.41067367792129517},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3880046010017395},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3647017776966095},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3380948305130005},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.33699384331703186},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.08374178409576416}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8214662075042725},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.7220975756645203},{"id":"https://openalex.org/C627467","wikidata":"https://www.wikidata.org/wiki/Q599027","display_name":"Harmonic function","level":2,"score":0.6925757527351379},{"id":"https://openalex.org/C2779560616","wikidata":"https://www.wikidata.org/wiki/Q1192869","display_name":"Dirichlet problem","level":3,"score":0.45065802335739136},{"id":"https://openalex.org/C127934551","wikidata":"https://www.wikidata.org/wiki/Q1148098","display_name":"Harmonic","level":2,"score":0.41067367792129517},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3880046010017395},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3647017776966095},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3380948305130005},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.33699384331703186},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.08374178409576416},{"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/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.0}],"mesh":[],"locations_count":7,"locations":[{"id":"doi:10.1007/s00454-019-00057-2","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00454-019-00057-2","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00454-019-00057-2.pdf","source":{"id":"https://openalex.org/S103671469","display_name":"Discrete & Computational Geometry","issn_l":"0179-5376","issn":["0179-5376","1432-0444"],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Discrete & Computational Geometry","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1707.07751","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1707.07751","pdf_url":"https://arxiv.org/pdf/1707.07751","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":"","raw_type":"text"},{"id":"pmh:oai:authors.library.caltech.edu:111007","is_oa":true,"landing_page_url":"https://authors.library.caltech.edu/111007/","pdf_url":"https://resolver.caltech.edu/CaltechAUTHORS:20210922-193309575","source":{"id":"https://openalex.org/S4306402161","display_name":"CaltechAUTHORS (California Institute of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I122411786","host_organization_name":"California Institute of Technology","host_organization_lineage":["https://openalex.org/I122411786"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article"},{"id":"mag:2736869835","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/1707.07751.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},{"id":"pmh:oai:www.repository.cam.ac.uk:1810/288814","is_oa":false,"landing_page_url":"https://www.repository.cam.ac.uk/handle/1810/288814","pdf_url":null,"source":{"id":"https://openalex.org/S4306401777","display_name":"Apollo (University of Cambridge)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I241749","host_organization_name":"University of Cambridge","host_organization_lineage":["https://openalex.org/I241749"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"},{"id":"doi:10.48550/arxiv.1707.07751","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1707.07751","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"},{"id":"doi:10.17863/cam.36077","is_oa":true,"landing_page_url":"https://doi.org/10.17863/cam.36077","pdf_url":null,"source":{"id":"https://openalex.org/S7407050737","display_name":"Apollo","issn_l":null,"issn":[],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"doi:10.1007/s00454-019-00057-2","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00454-019-00057-2","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00454-019-00057-2.pdf","source":{"id":"https://openalex.org/S103671469","display_name":"Discrete & Computational Geometry","issn_l":"0179-5376","issn":["0179-5376","1432-0444"],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Discrete & Computational Geometry","raw_type":"journal-article"},"sustainable_development_goals":[{"display_name":"Affordable and clean energy","score":0.5099999904632568,"id":"https://metadata.un.org/sdg/7"}],"awards":[],"funders":[{"id":"https://openalex.org/F4320308943","display_name":"Microsoft Research","ror":"https://ror.org/00d0nc645"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2736869835.pdf","grobid_xml":"https://content.openalex.org/works/W2736869835.grobid-xml"},"referenced_works_count":39,"referenced_works":["https://openalex.org/W121824094","https://openalex.org/W175559256","https://openalex.org/W194188155","https://openalex.org/W599944363","https://openalex.org/W1124258527","https://openalex.org/W1514389406","https://openalex.org/W1530501787","https://openalex.org/W1585244820","https://openalex.org/W1784898840","https://openalex.org/W1925985711","https://openalex.org/W1971788517","https://openalex.org/W2002975940","https://openalex.org/W2005142804","https://openalex.org/W2005704408","https://openalex.org/W2010968963","https://openalex.org/W2019892225","https://openalex.org/W2023760124","https://openalex.org/W2028212485","https://openalex.org/W2032573808","https://openalex.org/W2035223288","https://openalex.org/W2039280211","https://openalex.org/W2058210605","https://openalex.org/W2095434817","https://openalex.org/W2129652096","https://openalex.org/W2157756416","https://openalex.org/W2211003671","https://openalex.org/W2226627677","https://openalex.org/W2302690465","https://openalex.org/W2325550648","https://openalex.org/W2326411585","https://openalex.org/W2327392001","https://openalex.org/W2949699443","https://openalex.org/W2962754888","https://openalex.org/W2963274212","https://openalex.org/W2964205487","https://openalex.org/W2973743969","https://openalex.org/W3103751552","https://openalex.org/W4244414651","https://openalex.org/W4253970038"],"related_works":["https://openalex.org/W1521997389","https://openalex.org/W2949241598","https://openalex.org/W2004706092","https://openalex.org/W2061498803","https://openalex.org/W1484611276","https://openalex.org/W3081516709","https://openalex.org/W3082645745","https://openalex.org/W2963966866","https://openalex.org/W2021044869","https://openalex.org/W615259851","https://openalex.org/W2170653373","https://openalex.org/W2952300194","https://openalex.org/W3100597064","https://openalex.org/W1982924824","https://openalex.org/W3186063374","https://openalex.org/W1982766449","https://openalex.org/W3103351379","https://openalex.org/W3104664561","https://openalex.org/W2043422724","https://openalex.org/W2761848845"],"abstract_inverted_index":{"Benjamini":[0],"and":[1,44,71],"Schramm":[2],"(Invent":[3],"Math":[4],"126(3):565\u2013587,":[5],"1996)":[6],"used":[7],"circle":[8,46],"packing":[9,47],"to":[10],"prove":[11],"that":[12,34],"every":[13,36,45],"transient,":[14,37],"bounded":[15,38,59],"degree":[16],"planar":[17,41],"graph":[18],"admits":[19],"non-constant":[20],"harmonic":[21,66,75],"functions":[22,68,77],"of":[23,48,65,74],"finite":[24],"Dirichlet":[25,67,76],"energy.":[26],"We":[27],"refine":[28],"their":[29],"result,":[30],"showing":[31],"in":[32,50],"particular":[33],"for":[35],"degree,":[39],"simple":[40],"triangulation":[42],"T":[43,49,70],"a":[51,56],"domain":[52],"D,":[53],"there":[54],"is":[55],"canonical,":[57],"explicit":[58],"linear":[60],"isomorphism":[61],"between":[62],"the":[63,72],"space":[64,73],"on":[69,78],"D.":[79]},"counts_by_year":[{"year":2019,"cited_by_count":1}],"updated_date":"2026-03-27T05:58:40.876381","created_date":"2025-10-10T00:00:00"}