{"id":"https://openalex.org/W7220992121","doi":"https://doi.org/10.48550/arxiv.2610.10505","title":"The circle packing and Riemann uniformization embedding of the tree-weighted planar maps converges to Liouville quantum gravity","display_name":"The circle packing and Riemann uniformization embedding of the tree-weighted planar maps converges to Liouville quantum gravity","publication_year":2026,"publication_date":"2026-10-07","ids":{"openalex":"https://openalex.org/W7220992121","doi":"https://doi.org/10.48550/arxiv.2610.10505"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.10505","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.10505","pdf_url":"https://arxiv.org/pdf/2610.10505","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.10505","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5034106138","display_name":"Nina Holden","orcid":"https://orcid.org/0000-0001-9259-4704","observed_orcids":["https://orcid.org/0000-0001-9259-4704"]},"institutions":[{"id":"https://openalex.org/I57206974","display_name":"New York University","ror":"https://ror.org/0190ak572","country_code":"US","type":"education","lineage":["https://openalex.org/I57206974"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Holden, Nina","raw_affiliation_strings":["Courant Institute , New York University ,"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Courant Institute , New York University ,","institution_ids":["https://openalex.org/I57206974"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5010354424","display_name":"Pu Yu","orcid":"https://orcid.org/0000-0002-1897-3826","observed_orcids":["https://orcid.org/0000-0002-1897-3826"]},"institutions":[{"id":"https://openalex.org/I57206974","display_name":"New York University","ror":"https://ror.org/0190ak572","country_code":"US","type":"education","lineage":["https://openalex.org/I57206974"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Yu, Pu","raw_affiliation_strings":["Courant Institute , New York University ,"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Courant Institute , New York University ,","institution_ids":["https://openalex.org/I57206974"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I57206974"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.9667999744415283,"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":0.9667999744415283,"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/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.01769999973475933,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.0017000000225380063,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/circle-packing","display_name":"circle packing","score":0.7670000195503235},{"id":"https://openalex.org/keywords/random-planar-maps","display_name":"random planar maps","score":0.9049999713897705},{"id":"https://openalex.org/keywords/liouville-quantum-gravity","display_name":"Liouville quantum gravity","score":0.9909999966621399},{"id":"https://openalex.org/keywords/scaling-limits","display_name":"scaling limits","score":0.625},{"id":"https://openalex.org/keywords/schramm-loewner-evolution","display_name":"Schramm-Loewner evolution","score":0.8379999995231628}],"concepts":[{"id":"https://openalex.org/C149260544","wikidata":"https://www.wikidata.org/wiki/Q7885142","display_name":"Uniformization (probability theory)","level":5,"score":0.8389000296592712},{"id":"https://openalex.org/C26020477","wikidata":"https://www.wikidata.org/wiki/Q2109761","display_name":"Uniformization theorem","level":5,"score":0.8111000061035156},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.763700008392334},{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.7333999872207642},{"id":"https://openalex.org/C134786449","wikidata":"https://www.wikidata.org/wiki/Q3391255","display_name":"Planar","level":2,"score":0.6891000270843506},{"id":"https://openalex.org/C123115066","wikidata":"https://www.wikidata.org/wiki/Q5121501","display_name":"Circle packing","level":2,"score":0.579800009727478},{"id":"https://openalex.org/C199479865","wikidata":"https://www.wikidata.org/wiki/Q205966","display_name":"Riemann hypothesis","level":2,"score":0.5561000108718872},{"id":"https://openalex.org/C122044880","wikidata":"https://www.wikidata.org/wiki/Q5498822","display_name":"Ergodic theory","level":2,"score":0.5386999845504761},{"id":"https://openalex.org/C18556879","wikidata":"https://www.wikidata.org/wiki/Q753035","display_name":"Riemann surface","level":2,"score":0.527999997138977},{"id":"https://openalex.org/C112468886","wikidata":"https://www.wikidata.org/wiki/Q5535485","display_name":"Geometric function theory","level":3,"score":0.5277000069618225},{"id":"https://openalex.org/C154483964","wikidata":"https://www.wikidata.org/wiki/Q825857","display_name":"Riemann sphere","level":3,"score":0.4982999861240387},{"id":"https://openalex.org/C113174947","wikidata":"https://www.wikidata.org/wiki/Q2859736","display_name":"Tree (set theory)","level":2,"score":0.44269999861717224},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.43779999017715454},{"id":"https://openalex.org/C98214594","wikidata":"https://www.wikidata.org/wiki/Q850275","display_name":"Conformal map","level":2,"score":0.4302000105381012},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.413100004196167},{"id":"https://openalex.org/C101837359","wikidata":"https://www.wikidata.org/wiki/Q547823","display_name":"Planar graph","level":3,"score":0.3653999865055084},{"id":"https://openalex.org/C137618395","wikidata":"https://www.wikidata.org/wiki/Q576728","display_name":"Winding number","level":2,"score":0.3625999987125397},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3589000105857849},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.31349998712539673},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.31310001015663147},{"id":"https://openalex.org/C2777735758","wikidata":"https://www.wikidata.org/wiki/Q817765","display_name":"Path (computing)","level":2,"score":0.30790001153945923},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.29019999504089355},{"id":"https://openalex.org/C2781228216","wikidata":"https://www.wikidata.org/wiki/Q753035","display_name":"Compact Riemann surface","level":3,"score":0.275299996137619},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.263700008392334},{"id":"https://openalex.org/C150936888","wikidata":"https://www.wikidata.org/wiki/Q4769748","display_name":"Annulus (botany)","level":2,"score":0.2614000141620636},{"id":"https://openalex.org/C18762648","wikidata":"https://www.wikidata.org/wiki/Q42213","display_name":"Work (physics)","level":2,"score":0.26100000739097595},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.25619998574256897},{"id":"https://openalex.org/C36503486","wikidata":"https://www.wikidata.org/wiki/Q11235244","display_name":"Domain (mathematical analysis)","level":2,"score":0.2551000118255615},{"id":"https://openalex.org/C162329640","wikidata":"https://www.wikidata.org/wiki/Q332648","display_name":"Covering space","level":2,"score":0.2524999976158142},{"id":"https://openalex.org/C2776799497","wikidata":"https://www.wikidata.org/wiki/Q484298","display_name":"Surface (topology)","level":2,"score":0.2515999972820282}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.10505","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.10505","pdf_url":"https://arxiv.org/pdf/2610.10505","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2610.10505","doi":"https://doi.org/10.48550/arxiv.2610.10505","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.10505","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.10505","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.10505","pdf_url":"https://arxiv.org/pdf/2610.10505","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7220992121.pdf","grobid_xml":"https://content.openalex.org/works/W7220992121.grobid-xml"},"referenced_works_count":2,"referenced_works":["https://openalex.org/W2401672900","https://openalex.org/W7134267811"],"related_works":["https://openalex.org/W7220971231","https://openalex.org/W7221043987","https://openalex.org/W2619961998","https://openalex.org/W4415953534","https://openalex.org/W2810821118","https://openalex.org/W2964325867","https://openalex.org/W7134267811","https://openalex.org/W1594909327","https://openalex.org/W4410525942","https://openalex.org/W3012702702"],"abstract_inverted_index":{"We":[0],"prove":[1,46],"that":[2,47],"in":[3,81,89,92],"the":[4,31,36,48,54,97,106],"disk,":[5,19],"sphere,":[6],"whole-plane":[7],"topology,":[8],"spanning":[9],"tree":[10],"weighted":[11],"planar":[12,57,79,101],"maps":[13,58,80,102],"converge":[14,59],"to":[15,39,60,103],"$\\sqrt{2}$-Liouville":[16],"quantum":[17],"gravity":[18],"sphere":[20],"or":[21],"cone":[22],"under":[23],"circle":[24,71,87],"packing":[25,72],"and":[26,73,96],"Riemann":[27,74],"uniformization":[28,75],"embedding":[29,76],"as":[30],"number":[32],"of":[33,35,53,86,99],"faces":[34,52],"map":[37],"goes":[38],"infinity.":[40],"As":[41],"a":[42,93],"byproduct,":[43],"we":[44],"also":[45],"natural":[49],"path":[50],"on":[51,66,70],"embedded":[55],"tree-weighted":[56,100],"SLE$_8$.":[61],"The":[62],"proof":[63],"is":[64],"based":[65],"our":[67],"earlier":[68],"work":[69],"for":[77],"random":[78],"ergodic":[82],"scale-free":[83],"environments,":[84],"comparisons":[85],"packings":[88],"different":[90],"domains":[91],"companion":[94],"paper,":[95],"convergence":[98],"$\\sqrt{2}$-LQG":[104],"by":[105],"first":[107],"author.":[108]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-09T00:00:00"}
