{"id":"https://openalex.org/W7204929427","doi":"https://doi.org/10.48550/arxiv.2608.29425","title":"Analogue of Cardy's formula on the 60-degree parallelogram: a modular approach","display_name":"Analogue of Cardy's formula on the 60-degree parallelogram: a modular approach","publication_year":2026,"publication_date":"2026-08-29","ids":{"openalex":"https://openalex.org/W7204929427","doi":"https://doi.org/10.48550/arxiv.2608.29425"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.29425","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.29425","pdf_url":"https://arxiv.org/pdf/2608.29425","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/2608.29425","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5148906842","display_name":"Cody R. Strouse","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Strouse, Cody R.","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5148906842"],"corresponding_institution_ids":[],"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/T11428","display_name":"Advanced Mathematical Identities","score":0.6265000104904175,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/T11428","display_name":"Advanced Mathematical Identities","score":0.6265000104904175,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/T11822","display_name":"Analytic and geometric function theory","score":0.10530000180006027,"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/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.05130000039935112,"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/critical-percolation","display_name":"critical percolation","score":0.9020000100135803},{"id":"https://openalex.org/keywords/crossing-probabilities","display_name":"crossing probabilities","score":0.7770000100135803},{"id":"https://openalex.org/keywords/scaling-limits","display_name":"scaling limits","score":0.5669999718666077},{"id":"https://openalex.org/keywords/modular-forms","display_name":"modular forms","score":0.5680000185966492},{"id":"https://openalex.org/keywords/conformal-symmetry","display_name":"conformal invariance","score":0.7229999899864197},{"id":"https://openalex.org/keywords/triangular-lattice","display_name":"triangular lattice","score":0.6200000047683716},{"id":"https://openalex.org/keywords/parallelogram","display_name":"parallelogram","score":0.5559999942779541},{"id":"https://openalex.org/keywords/eta-function","display_name":"eta function","score":0.6970000267028809},{"id":"https://openalex.org/keywords/uniqueness-theorem","display_name":"uniqueness theorem","score":0.7900000214576721},{"id":"https://openalex.org/keywords/q-expansions","display_name":"q-expansions","score":0.9179999828338623}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8129000067710876},{"id":"https://openalex.org/C2781302577","wikidata":"https://www.wikidata.org/wiki/Q209","display_name":"Rectangle","level":2,"score":0.6064000129699707},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.5946000218391418},{"id":"https://openalex.org/C101468663","wikidata":"https://www.wikidata.org/wiki/Q1620158","display_name":"Modular design","level":2,"score":0.5609999895095825},{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.48910000920295715},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.47780001163482666},{"id":"https://openalex.org/C130979935","wikidata":"https://www.wikidata.org/wiki/Q568954","display_name":"Ansatz","level":2,"score":0.47099998593330383},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.47040000557899475},{"id":"https://openalex.org/C75764964","wikidata":"https://www.wikidata.org/wiki/Q870797","display_name":"Modular form","level":2,"score":0.4641999900341034},{"id":"https://openalex.org/C36503486","wikidata":"https://www.wikidata.org/wiki/Q11235244","display_name":"Domain (mathematical analysis)","level":2,"score":0.4571000039577484},{"id":"https://openalex.org/C98214594","wikidata":"https://www.wikidata.org/wiki/Q850275","display_name":"Conformal map","level":2,"score":0.45249998569488525},{"id":"https://openalex.org/C153576954","wikidata":"https://www.wikidata.org/wiki/Q594452","display_name":"Functional equation","level":3,"score":0.413100004196167},{"id":"https://openalex.org/C97900337","wikidata":"https://www.wikidata.org/wiki/Q60367","display_name":"Modular invariance","level":3,"score":0.4090000092983246},{"id":"https://openalex.org/C99844830","wikidata":"https://www.wikidata.org/wiki/Q102441924","display_name":"Scaling","level":2,"score":0.40369999408721924},{"id":"https://openalex.org/C204911207","wikidata":"https://www.wikidata.org/wiki/Q860609","display_name":"Generating function","level":2,"score":0.39989998936653137},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3747999966144562},{"id":"https://openalex.org/C105546189","wikidata":"https://www.wikidata.org/wiki/Q1154787","display_name":"Theta function","level":2,"score":0.3700999915599823},{"id":"https://openalex.org/C59766168","wikidata":"https://www.wikidata.org/wiki/Q3001220","display_name":"Modular curve","level":3,"score":0.35339999198913574},{"id":"https://openalex.org/C24084028","wikidata":"https://www.wikidata.org/wiki/Q7429781","display_name":"Scaling limit","level":3,"score":0.33410000801086426},{"id":"https://openalex.org/C105462591","wikidata":"https://www.wikidata.org/wiki/Q3737844","display_name":"Binary function","level":3,"score":0.33070001006126404},{"id":"https://openalex.org/C54613284","wikidata":"https://www.wikidata.org/wiki/Q1191271","display_name":"Conformal field theory","level":3,"score":0.30320000648498535},{"id":"https://openalex.org/C173848574","wikidata":"https://www.wikidata.org/wiki/Q3507963","display_name":"Conformal symmetry","level":3,"score":0.29980000853538513},{"id":"https://openalex.org/C134786449","wikidata":"https://www.wikidata.org/wiki/Q3391255","display_name":"Planar","level":2,"score":0.29339998960494995},{"id":"https://openalex.org/C97937538","wikidata":"https://www.wikidata.org/wiki/Q199691","display_name":"Laplace transform","level":2,"score":0.2754000127315521},{"id":"https://openalex.org/C2780457167","wikidata":"https://www.wikidata.org/wiki/Q17156484","display_name":"Percolation (cognitive psychology)","level":2,"score":0.2703999876976013},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.26489999890327454},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.26109999418258667},{"id":"https://openalex.org/C2778067643","wikidata":"https://www.wikidata.org/wiki/Q166507","display_name":"Interval (graph theory)","level":2,"score":0.2549000084400177},{"id":"https://openalex.org/C165700671","wikidata":"https://www.wikidata.org/wiki/Q203484","display_name":"Laplace operator","level":2,"score":0.2529999911785126}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.29425","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.29425","pdf_url":"https://arxiv.org/pdf/2608.29425","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.2608.29425","doi":"https://doi.org/10.48550/arxiv.2608.29425","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.29425","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"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2608.29425","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.29425","pdf_url":"https://arxiv.org/pdf/2608.29425","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":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7204929427.pdf","grobid_xml":"https://content.openalex.org/works/W7204929427.grobid-xml"},"referenced_works_count":4,"referenced_works":["https://openalex.org/W1528818655","https://openalex.org/W1559976006","https://openalex.org/W1797893441","https://openalex.org/W2041250014"],"related_works":["https://openalex.org/W7201212040","https://openalex.org/W1553493265","https://openalex.org/W1528818655","https://openalex.org/W7095800588","https://openalex.org/W7118164865","https://openalex.org/W4386648660","https://openalex.org/W2963283280","https://openalex.org/W2156895775","https://openalex.org/W1551131215","https://openalex.org/W4409289111"],"abstract_inverted_index":{"The":[0],"Cardy-Smirnov":[1,103],"function":[2,67,104,147],"of":[3,13,34,120],"a":[4,22,69,76,82,97,111,127,132,138,142],"four-marked":[5],"planar":[6],"domain":[7],"encodes":[8],"the":[9,14,29,32,43,54,64,66,89,92,102,121,146],"conjectural":[10],"scaling":[11],"limit":[12,27],"crossing":[15],"probability":[16],"for":[17,25,49,101],"critical":[18,50],"percolation.":[19],"Cardy":[20],"predicted":[21],"closed":[23,98,112],"formula":[24,44,100,114],"this":[26],"on":[28,53,63,91],"rectangle":[30,65],"in":[31],"cases":[33],"both":[35],"site":[36,51],"and":[37,40,58,125,141],"bond":[38],"percolation,":[39],"Smirnov":[41],"proved":[42],"together":[45,80,136],"with":[46,81,137],"conformal":[47],"invariance":[48],"percolation":[52],"triangular":[55],"lattice.":[56],"Kleban":[57],"Zagier":[59],"later":[60],"showed":[61],"that":[62,131],"admits":[68],"modular":[70,77,113,122],"interpretation:":[71],"it":[72,116],"is":[73],"determined":[74],"by":[75],"functional":[78,134],"equation":[79],"mild":[83],"analytic":[84],"ansatz.":[85],"We":[86,95],"carry":[87],"out":[88],"analogue":[90],"$\u03c0/3$":[93],"parallelogram.":[94],"derive":[96],"conformal-map":[99],"as":[105,117],"an":[106,118],"incomplete":[107],"beta":[108],"integral,":[109],"prove":[110],"realizing":[115],"integral":[119],"form":[123],"$\u03b7(\u03c4)^2\u03b7(3\u03c4)^2$,":[124],"establish":[126],"uniqueness":[128],"theorem":[129],"showing":[130],"single":[133],"equation,":[135],"$q$-expansion":[139],"ansatz":[140],"nondegeneracy":[143],"condition,":[144],"determines":[145],"uniquely.":[148]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-02T00:00:00"}
