{"id":"https://openalex.org/W7204789476","doi":"https://doi.org/10.48550/arxiv.2608.28328","title":"Critical Gaussian Multiplicative Chaos on the Circle Is Rajchman","display_name":"Critical Gaussian Multiplicative Chaos on the Circle Is Rajchman","publication_year":2026,"publication_date":"2026-08-28","ids":{"openalex":"https://openalex.org/W7204789476","doi":"https://doi.org/10.48550/arxiv.2608.28328"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.28328","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.28328","pdf_url":"https://arxiv.org/pdf/2608.28328","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/2608.28328","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5100430546","display_name":"Yin Cai","orcid":"https://orcid.org/0000-0001-5260-7515","observed_orcids":["https://orcid.org/0000-0001-5260-7515"]},"institutions":[{"id":"https://openalex.org/I9842412","display_name":"Nanjing University of Aeronautics and Astronautics","ror":"https://ror.org/01scyh794","country_code":"CN","type":"education","lineage":["https://openalex.org/I9842412"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Cai, Yin","raw_affiliation_strings":["School of Mathematics , Nanjing University of Aeronautics and Astronautics , Nanjing 210016 , P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics , Nanjing University of Aeronautics and Astronautics , Nanjing 210016 , P. R. China","institution_ids":["https://openalex.org/I9842412"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5148819924","display_name":"Bonan Chen","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I9842412","display_name":"Nanjing University of Aeronautics and Astronautics","ror":"https://ror.org/01scyh794","country_code":"CN","type":"education","lineage":["https://openalex.org/I9842412"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Chen, Bonan","raw_affiliation_strings":["School of Mathematics , Nanjing University of Aeronautics and Astronautics , Nanjing 210016 , P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics , Nanjing University of Aeronautics and Astronautics , Nanjing 210016 , P. R. China","institution_ids":["https://openalex.org/I9842412"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5148831483","display_name":"Xiang Fang","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I9842412","display_name":"Nanjing University of Aeronautics and Astronautics","ror":"https://ror.org/01scyh794","country_code":"CN","type":"education","lineage":["https://openalex.org/I9842412"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Fang, Xiang","raw_affiliation_strings":["School of Mathematics , Nanjing University of Aeronautics and Astronautics , Nanjing 210016 , P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics , Nanjing University of Aeronautics and Astronautics , Nanjing 210016 , P. R. China","institution_ids":["https://openalex.org/I9842412"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5151745328","display_name":"Feng Guo","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I9842412","display_name":"Nanjing University of Aeronautics and Astronautics","ror":"https://ror.org/01scyh794","country_code":"CN","type":"education","lineage":["https://openalex.org/I9842412"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Guo, Feng","raw_affiliation_strings":["School of Mathematics , Nanjing University of Aeronautics and Astronautics , Nanjing 210016 , P. R. China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mathematics , Nanjing University of Aeronautics and Astronautics , Nanjing 210016 , P. R. China","institution_ids":["https://openalex.org/I9842412"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I9842412"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":4,"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":0.6542999744415283,"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":0.6542999744415283,"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/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.08659999817609787,"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/T13234","display_name":"p-Adic Analysis and Mathematical Physics","score":0.0697999969124794,"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/fourier-coefficients","display_name":"Fourier coefficients","score":0.8970000147819519},{"id":"https://openalex.org/keywords/fourier-dimension","display_name":"Fourier dimension","score":0.5590000152587891},{"id":"https://openalex.org/keywords/covariance-correction","display_name":"covariance correction","score":0.6579999923706055}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.829200029373169},{"id":"https://openalex.org/C42747912","wikidata":"https://www.wikidata.org/wiki/Q1048447","display_name":"Multiplicative function","level":2,"score":0.5781999826431274},{"id":"https://openalex.org/C207864730","wikidata":"https://www.wikidata.org/wiki/Q179467","display_name":"Fourier series","level":2,"score":0.5647000074386597},{"id":"https://openalex.org/C178650346","wikidata":"https://www.wikidata.org/wiki/Q201984","display_name":"Covariance","level":2,"score":0.5241000056266785},{"id":"https://openalex.org/C102519508","wikidata":"https://www.wikidata.org/wiki/Q6520159","display_name":"Fourier transform","level":2,"score":0.46070000529289246},{"id":"https://openalex.org/C203024314","wikidata":"https://www.wikidata.org/wiki/Q1365258","display_name":"Fourier analysis","level":3,"score":0.4602999985218048},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.44339999556541443},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.3828999996185303},{"id":"https://openalex.org/C126875415","wikidata":"https://www.wikidata.org/wiki/Q203425","display_name":"Unit circle","level":2,"score":0.3626999855041504},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.33709999918937683},{"id":"https://openalex.org/C137250428","wikidata":"https://www.wikidata.org/wiki/Q5178897","display_name":"Covariance function","level":3,"score":0.3280999958515167},{"id":"https://openalex.org/C2777021972","wikidata":"https://www.wikidata.org/wiki/Q22976830","display_name":"Uniqueness","level":2,"score":0.3197000026702881},{"id":"https://openalex.org/C175225751","wikidata":"https://www.wikidata.org/wiki/Q15927242","display_name":"Discrete Fourier series","level":5,"score":0.31949999928474426},{"id":"https://openalex.org/C61326573","wikidata":"https://www.wikidata.org/wiki/Q1496376","display_name":"Gaussian process","level":3,"score":0.31360000371932983},{"id":"https://openalex.org/C2779374083","wikidata":"https://www.wikidata.org/wiki/Q5011038","display_name":"CHAOS (operating system)","level":2,"score":0.28220000863075256},{"id":"https://openalex.org/C112633086","wikidata":"https://www.wikidata.org/wiki/Q381287","display_name":"White noise","level":2,"score":0.27720001339912415},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.2703999876976013},{"id":"https://openalex.org/C197656079","wikidata":"https://www.wikidata.org/wiki/Q17147719","display_name":"Polynomial chaos","level":3,"score":0.26269999146461487},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.2535000145435333}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.28328","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.28328","pdf_url":"https://arxiv.org/pdf/2608.28328","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.2608.28328","doi":"https://doi.org/10.48550/arxiv.2608.28328","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.28328","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:2608.28328","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.28328","pdf_url":"https://arxiv.org/pdf/2608.28328","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/W7204789476.pdf","grobid_xml":"https://content.openalex.org/works/W7204789476.grobid-xml"},"referenced_works_count":7,"referenced_works":["https://openalex.org/W1973766190","https://openalex.org/W2022385268","https://openalex.org/W2583433875","https://openalex.org/W2888362840","https://openalex.org/W4388514831","https://openalex.org/W7102390685","https://openalex.org/W7169760095"],"related_works":["https://openalex.org/W7220731556","https://openalex.org/W4388514831","https://openalex.org/W7102390685","https://openalex.org/W7117144399","https://openalex.org/W4404650254","https://openalex.org/W4297834112","https://openalex.org/W4417252704","https://openalex.org/W7212406441","https://openalex.org/W2583433875","https://openalex.org/W4414920969"],"abstract_inverted_index":{"We":[0],"prove":[1],"that":[2],"the":[3,7,14,26,32,52,58,78,86,98,107,147,160,187,191,196],"Fourier":[4,66,83,90,148,162],"coefficients":[5,163],"of":[6,37,88,103,136,159,195],"canonical":[8,27,59,192],"critical":[9,28,54,63,104,193],"Gaussian":[10],"multiplicative":[11],"chaos":[12,29,64,105,194],"on":[13],"circle":[15,34,60,198],"vanish":[16],"almost":[17,44,69],"surely":[18],"at":[19,142],"infinity.":[20],"More":[21],"precisely,":[22],"let":[23],"$M_\u03d5^{\\mathrm{crit}}$":[24],"be":[25],"associated":[30],"with":[31],"centered":[33],"field":[35],"$\u03d5$":[36],"covariance":[38,180],"$\\mathbb{E}[\u03d5(\u03b8)\u03d5(\u03b8')]":[39],"=":[40],"\\log\\frac{1}{\\lvert":[41],"e^{i\u03b8}-e^{i\u03b8'}\\rvert}$.":[42],"Then,":[43],"surely,":[45,70],"$\\widehat{M_\u03d5^{\\mathrm{crit}}}(n)\\longrightarrow0$":[46],"as":[47],"$\\lvert":[48],"n\\rvert\\to\\infty$.":[49],"This":[50],"resolves":[51],"almost-sure":[53],"Rajchman":[55,188],"problem":[56],"for":[57,172],"field.":[61,199],"Since":[62],"has":[65],"dimension":[67],"zero":[68],"no":[71],"positive":[72,89],"polynomial":[73],"Fourier-decay":[74],"rate":[75],"can":[76],"hold;":[77],"theorem":[79],"therefore":[80],"exhibits":[81],"qualitative":[82],"cancellation":[84],"beyond":[85],"regime":[87],"dimension.":[91],"The":[92],"proof":[93],"addresses":[94],"two":[95],"coupled":[96],"difficulties:":[97],"heavy,":[99],"nonuniform":[100],"cell":[101],"masses":[102],"and":[106,133,152,182],"need":[108],"to":[109,168,190],"control":[110,135,158],"exponentially":[111],"many":[112],"frequencies":[113],"in":[114],"each":[115,165],"dyadic":[116,166],"annulus.":[117],"For":[118],"an":[119],"auxiliary":[120],"periodized":[121],"compact-range":[122],"star-scale":[123],"field,":[124],"a":[125,143,169,173],"derivative-rooted":[126],"Bessel":[127],"regression":[128],"yields":[129],"weighted":[130],"small-cell":[131],"summability":[132],"moving-tail":[134],"exceptional":[137],"large":[138],"cells.":[139],"After":[140],"conditioning":[141],"coarse":[144,174],"scale":[145],"below":[146],"scale,":[149],"finite-range":[150],"independence":[151],"conditional":[153],"Bernstein":[154],"concentration":[155],"reduce":[156],"uniform":[157],"terminal":[161],"over":[164],"annulus":[167],"spatial-variation":[170],"estimate":[171],"predictable":[175],"measure.":[176],"A":[177],"smooth":[178],"positive-definite":[179],"correction":[181],"critical-chaos":[183],"uniqueness":[184],"then":[185],"transfer":[186],"property":[189],"exact":[197]},"counts_by_year":[{"year":2026,"cited_by_count":4}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-01T00:00:00"}