{"id":"https://openalex.org/W7213331306","doi":"https://doi.org/10.48550/arxiv.2609.27861","title":"Closed Response Calculus and Joint Densities for the Liouville Quantum Gravity Metric","display_name":"Closed Response Calculus and Joint Densities for the Liouville Quantum Gravity Metric","publication_year":2026,"publication_date":"2026-08-20","ids":{"openalex":"https://openalex.org/W7213331306","doi":"https://doi.org/10.48550/arxiv.2609.27861"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.27861","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.27861","pdf_url":"https://arxiv.org/pdf/2609.27861","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/2609.27861","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5077898392","display_name":"Chunhao Cai","orcid":"https://orcid.org/0000-0001-7483-4479","observed_orcids":["https://orcid.org/0000-0001-7483-4479"]},"institutions":[{"id":"https://openalex.org/I157773358","display_name":"Sun Yat-sen University","ror":"https://ror.org/0064kty71","country_code":"CN","type":"education","lineage":["https://openalex.org/I157773358"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Cai, Chunhao","raw_affiliation_strings":["School of Mathematics (Zhuhai) , Sun Yat-Sen University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics (Zhuhai) , Sun Yat-Sen University","institution_ids":["https://openalex.org/I157773358"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5077898392"],"corresponding_institution_ids":["https://openalex.org/I157773358"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"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":0.7235999703407288,"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":0.7235999703407288,"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":0.25270000100135803,"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/T13234","display_name":"p-Adic Analysis and Mathematical Physics","score":0.003599999938160181,"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/joint-density","display_name":"joint density","score":0.6710000038146973},{"id":"https://openalex.org/keywords/integration-by-parts","display_name":"integration by parts","score":0.777999997138977},{"id":"https://openalex.org/keywords/divergence","display_name":"divergence","score":0.6129999756813049},{"id":"https://openalex.org/keywords/geodesics","display_name":"geodesics","score":0.8669999837875366},{"id":"https://openalex.org/keywords/lebesgue-density","display_name":"Lebesgue density","score":0.6190000176429749}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8163999915122986},{"id":"https://openalex.org/C165818556","wikidata":"https://www.wikidata.org/wiki/Q213488","display_name":"Geodesic","level":2,"score":0.7239000201225281},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5526999831199646},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.5266000032424927},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4575999975204468},{"id":"https://openalex.org/C204323151","wikidata":"https://www.wikidata.org/wiki/Q905424","display_name":"Range (aeronautics)","level":2,"score":0.40369999408721924},{"id":"https://openalex.org/C36649600","wikidata":"https://www.wikidata.org/wiki/Q762521","display_name":"Logarithmic derivative","level":2,"score":0.390500009059906},{"id":"https://openalex.org/C192939610","wikidata":"https://www.wikidata.org/wiki/Q188444","display_name":"Differential geometry","level":2,"score":0.38679999113082886},{"id":"https://openalex.org/C18648836","wikidata":"https://www.wikidata.org/wiki/Q381892","display_name":"Compact space","level":2,"score":0.37950000166893005},{"id":"https://openalex.org/C56802139","wikidata":"https://www.wikidata.org/wiki/Q149999","display_name":"Differential calculus","level":2,"score":0.37459999322891235},{"id":"https://openalex.org/C99730327","wikidata":"https://www.wikidata.org/wiki/Q1501536","display_name":"Sobolev space","level":2,"score":0.3659000098705292},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.3433000147342682},{"id":"https://openalex.org/C118365987","wikidata":"https://www.wikidata.org/wiki/Q273328","display_name":"Integration by parts","level":2,"score":0.33880001306533813},{"id":"https://openalex.org/C98763669","wikidata":"https://www.wikidata.org/wiki/Q176645","display_name":"Markov chain","level":2,"score":0.3321000039577484},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.32899999618530273},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.3093999922275543},{"id":"https://openalex.org/C198043062","wikidata":"https://www.wikidata.org/wiki/Q180953","display_name":"Metric space","level":2,"score":0.3061000108718872},{"id":"https://openalex.org/C2778029271","wikidata":"https://www.wikidata.org/wiki/Q5421931","display_name":"Extension (predicate logic)","level":2,"score":0.30559998750686646},{"id":"https://openalex.org/C49766605","wikidata":"https://www.wikidata.org/wiki/Q207643","display_name":"Linear map","level":2,"score":0.3052999973297119},{"id":"https://openalex.org/C4306028","wikidata":"https://www.wikidata.org/wiki/Q2185349","display_name":"Fundamental theorem of Riemannian geometry","level":4,"score":0.29789999127388},{"id":"https://openalex.org/C2779593128","wikidata":"https://www.wikidata.org/wiki/Q632814","display_name":"Riemannian manifold","level":2,"score":0.29319998621940613},{"id":"https://openalex.org/C2777105136","wikidata":"https://www.wikidata.org/wiki/Q827230","display_name":"Lebesgue measure","level":3,"score":0.29019999504089355},{"id":"https://openalex.org/C82457910","wikidata":"https://www.wikidata.org/wiki/Q740207","display_name":"Isometry (Riemannian geometry)","level":2,"score":0.28999999165534973},{"id":"https://openalex.org/C19492387","wikidata":"https://www.wikidata.org/wiki/Q2287715","display_name":"Covariant derivative","level":2,"score":0.28929999470710754},{"id":"https://openalex.org/C112313634","wikidata":"https://www.wikidata.org/wiki/Q7886648","display_name":"Complement (music)","level":5,"score":0.2867000102996826},{"id":"https://openalex.org/C2776716506","wikidata":"https://www.wikidata.org/wiki/Q5161158","display_name":"Conjugate points","level":2,"score":0.2849000096321106},{"id":"https://openalex.org/C204241405","wikidata":"https://www.wikidata.org/wiki/Q461499","display_name":"Transformation (genetics)","level":3,"score":0.28459998965263367},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.2800000011920929},{"id":"https://openalex.org/C2777630202","wikidata":"https://www.wikidata.org/wiki/Q5527840","display_name":"Gaussian measure","level":3,"score":0.2791999876499176},{"id":"https://openalex.org/C14158598","wikidata":"https://www.wikidata.org/wiki/Q853831","display_name":"Lebesgue integration","level":2,"score":0.2743000090122223},{"id":"https://openalex.org/C111771559","wikidata":"https://www.wikidata.org/wiki/Q66295","display_name":"Derivative (finance)","level":2,"score":0.2736000120639801},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.2732999920845032},{"id":"https://openalex.org/C2780791683","wikidata":"https://www.wikidata.org/wiki/Q846785","display_name":"Action (physics)","level":2,"score":0.2718999981880188},{"id":"https://openalex.org/C102224218","wikidata":"https://www.wikidata.org/wiki/Q25344559","display_name":"Exponential map (Riemannian geometry)","level":5,"score":0.2711000144481659},{"id":"https://openalex.org/C108568745","wikidata":"https://www.wikidata.org/wiki/Q234181","display_name":"Quantum gravity","level":3,"score":0.2709999978542328},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.26980000734329224},{"id":"https://openalex.org/C21031990","wikidata":"https://www.wikidata.org/wiki/Q355020","display_name":"Probability measure","level":2,"score":0.2630999982357025},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.2605000138282776},{"id":"https://openalex.org/C17137986","wikidata":"https://www.wikidata.org/wiki/Q215067","display_name":"Orthogonality","level":2,"score":0.26010000705718994},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.257099986076355}],"mesh":[],"locations_count":4,"locations":[{"id":"pmh:oai:arXiv.org:2609.27861","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.27861","pdf_url":"https://arxiv.org/pdf/2609.27861","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.2609.27861","doi":"https://doi.org/10.48550/arxiv.2609.27861","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.27861","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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},{"id":"doi:10.5281/zenodo.22772246","doi":"https://doi.org/10.5281/zenodo.22772246","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.22772246","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},{"id":"doi:10.5281/zenodo.22772247","doi":"https://doi.org/10.5281/zenodo.22772247","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.22772247","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.27861","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.27861","pdf_url":"https://arxiv.org/pdf/2609.27861","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/W7213331306.pdf","grobid_xml":"https://content.openalex.org/works/W7213331306.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7214388122","https://openalex.org/W3197597653","https://openalex.org/W4400163339","https://openalex.org/W6906746489","https://openalex.org/W7097657124","https://openalex.org/W4313641232","https://openalex.org/W3203863159","https://openalex.org/W4389775325","https://openalex.org/W4288362889","https://openalex.org/W7155204981"],"abstract_inverted_index":{"We":[0,43],"give":[1],"a":[2,15,33,39,67,97,110,136],"general":[3],"criterion":[4],"for":[5,120],"transferring":[6],"first-order":[7],"responses":[8],"on":[9,19,88],"an":[10,23],"underlying":[11],"probability":[12],"space":[13],"to":[14,47],"closed":[16,40],"differential":[17],"calculus":[18],"the":[20,48,55,64,72,77,89,94,103,114],"law":[21,92],"of":[22,60,66,76,79,96,124,130],"observable.":[24],"An":[25],"integration-by-parts":[26],"identity":[27],"removes":[28],"presentation":[29],"ambiguity":[30],"and":[31,38,101,109],"yields":[32],"closable":[34],"gradient,":[35],"its":[36],"divergence,":[37],"Markov":[41],"form.":[42],"apply":[44],"this":[45],"scheme":[46],"subcritical":[49],"Liouville":[50],"quantum":[51],"gravity":[52],"metric":[53,91],"throughout":[54],"range":[56],"$0<\u03b3<2$.":[57],"Sequential":[58],"compactness":[59],"Weyl-perturbed":[61],"geodesics":[62],"identifies":[63],"derivative":[65],"logarithmic":[68,131],"distance":[69,132],"ratio":[70],"with":[71],"Sobolev":[73],"Riesz":[74],"representative":[75],"difference":[78],"two":[80],"normalized":[81],"geodesic":[82],"occupation":[83],"measures.":[84],"The":[85,127],"induced":[86],"form":[87,100],"projective":[90],"is":[93],"image":[95],"Gaussian":[98],"directional":[99],"has":[102,135],"energy-image-density":[104],"property.":[105],"Finally,":[106],"fixed-target":[107],"confluence":[108],"leaf-elimination":[111],"argument":[112],"make":[113],"response":[115],"Gram":[116],"matrix":[117],"positive":[118],"definite":[119],"every":[121],"finite":[122],"forest":[123],"marked":[125],"pairs.":[126],"corresponding":[128],"vector":[129],"ratios":[133],"therefore":[134],"Lebesgue":[137],"density.":[138]},"counts_by_year":[],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2026-09-16T00:00:00"}