{"id":"https://openalex.org/W7212110906","doi":"https://doi.org/10.48550/arxiv.2609.09642","title":"Sharp Typical Distance and Exponential Small-Ball Bounds in One-Step-Cliff Nielsen Geometry","display_name":"Sharp Typical Distance and Exponential Small-Ball Bounds in One-Step-Cliff Nielsen Geometry","publication_year":2026,"publication_date":"2026-09-09","ids":{"openalex":"https://openalex.org/W7212110906","doi":"https://doi.org/10.48550/arxiv.2609.09642"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.09642","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.09642","pdf_url":"https://arxiv.org/pdf/2609.09642","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/2609.09642","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5030260834","display_name":"Honghuai Fang","orcid":"https://orcid.org/0009-0005-4455-2899","observed_orcids":["https://orcid.org/0009-0005-4455-2899"]},"institutions":[{"id":"https://openalex.org/I3133055985","display_name":"Westlake University","ror":"https://ror.org/05hfa4n20","country_code":"CN","type":"education","lineage":["https://openalex.org/I3133055985"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Fang, Honghuai","raw_affiliation_strings":["Institute for Theoretical Sciences , Westlake University , 600 Dunyu Road , Xihu District , Hangzhou , Zhejiang 310030 , China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute for Theoretical Sciences , Westlake University , 600 Dunyu Road , Xihu District , Hangzhou , Zhejiang 310030 , China","institution_ids":["https://openalex.org/I3133055985"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5030260834"],"corresponding_institution_ids":["https://openalex.org/I3133055985"],"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/T11716","display_name":"Random Matrices and Applications","score":0.23720000684261322,"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"}},"topics":[{"id":"https://openalex.org/T11716","display_name":"Random Matrices and Applications","score":0.23720000684261322,"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"}},{"id":"https://openalex.org/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.12770000100135803,"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/T11680","display_name":"Advanced Algebra and Geometry","score":0.09109999984502792,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/projective-unitary-group","display_name":"projective unitary group","score":0.7960000038146973},{"id":"https://openalex.org/keywords/haar-measure","display_name":"Haar measure","score":0.6669999957084656},{"id":"https://openalex.org/keywords/random-matrix-theory","display_name":"random matrix theory","score":0.41999998688697815},{"id":"https://openalex.org/keywords/small-ball-probabilities","display_name":"small ball probabilities","score":0.6110000014305115},{"id":"https://openalex.org/keywords/circular-unitary-ensemble","display_name":"circular unitary ensemble","score":0.7139999866485596},{"id":"https://openalex.org/keywords/logarithmic-energy","display_name":"logarithmic energy","score":0.6629999876022339}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.857699990272522},{"id":"https://openalex.org/C165818556","wikidata":"https://www.wikidata.org/wiki/Q213488","display_name":"Geodesic","level":2,"score":0.6531999707221985},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.6381000280380249},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5971999764442444},{"id":"https://openalex.org/C130805567","wikidata":"https://www.wikidata.org/wiki/Q1162676","display_name":"Haar measure","level":2,"score":0.5353999733924866},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.535099983215332},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.5074999928474426},{"id":"https://openalex.org/C112313634","wikidata":"https://www.wikidata.org/wiki/Q7886648","display_name":"Complement (music)","level":5,"score":0.49000000953674316},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.4253999888896942},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.42309999465942383},{"id":"https://openalex.org/C179254644","wikidata":"https://www.wikidata.org/wiki/Q13222844","display_name":"Moment (physics)","level":2,"score":0.41119998693466187},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.40560001134872437},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.39149999618530273},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3790999948978424},{"id":"https://openalex.org/C21031990","wikidata":"https://www.wikidata.org/wiki/Q355020","display_name":"Probability measure","level":2,"score":0.34850001335144043},{"id":"https://openalex.org/C2778067643","wikidata":"https://www.wikidata.org/wiki/Q166507","display_name":"Interval (graph theory)","level":2,"score":0.3424000144004822},{"id":"https://openalex.org/C94940","wikidata":"https://www.wikidata.org/wiki/Q652941","display_name":"Hermitian matrix","level":2,"score":0.30469998717308044},{"id":"https://openalex.org/C2778355321","wikidata":"https://www.wikidata.org/wiki/Q17079427","display_name":"Identity (music)","level":2,"score":0.30149999260902405},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.2939000129699707},{"id":"https://openalex.org/C182964748","wikidata":"https://www.wikidata.org/wiki/Q208216","display_name":"Triangle inequality","level":2,"score":0.29319998621940613},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.2874000072479248},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.2768000066280365},{"id":"https://openalex.org/C96214315","wikidata":"https://www.wikidata.org/wiki/Q727103","display_name":"Unitary matrix","level":3,"score":0.2678000032901764},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.25519999861717224},{"id":"https://openalex.org/C2777634741","wikidata":"https://www.wikidata.org/wiki/Q768993","display_name":"Wasserstein metric","level":2,"score":0.25440001487731934},{"id":"https://openalex.org/C67820243","wikidata":"https://www.wikidata.org/wiki/Q179164","display_name":"Unitary state","level":2,"score":0.25380000472068787},{"id":"https://openalex.org/C75291252","wikidata":"https://www.wikidata.org/wiki/Q1315756","display_name":"TRACE (psycholinguistics)","level":2,"score":0.2531999945640564},{"id":"https://openalex.org/C192939610","wikidata":"https://www.wikidata.org/wiki/Q188444","display_name":"Differential geometry","level":2,"score":0.2517000138759613}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.09642","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.09642","pdf_url":"https://arxiv.org/pdf/2609.09642","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.2609.09642","doi":"https://doi.org/10.48550/arxiv.2609.09642","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.09642","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:2609.09642","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.09642","pdf_url":"https://arxiv.org/pdf/2609.09642","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/W7212110906.pdf","grobid_xml":"https://content.openalex.org/works/W7212110906.grobid-xml"},"referenced_works_count":3,"referenced_works":["https://openalex.org/W4225592834","https://openalex.org/W4402337924","https://openalex.org/W7163365022"],"related_works":["https://openalex.org/W7208704283","https://openalex.org/W7098673386","https://openalex.org/W2563617918","https://openalex.org/W4392011817","https://openalex.org/W2151637719","https://openalex.org/W2962735967","https://openalex.org/W4404044951","https://openalex.org/W7008757102","https://openalex.org/W7042620228","https://openalex.org/W4377002742"],"abstract_inverted_index":{"Let":[0],"$D=2^n$":[1],"and":[2,21,23,49,93,120,142],"equip":[3],"$\\operatorname{PU}(D)$":[4],"with":[5,10],"the":[6,32,35,91,116,127,136,145,153],"one-step-cliff":[7],"Nielsen":[8],"metric,":[9],"quadratic":[11],"metric":[12],"coefficients":[13],"one":[14,20],"in":[15,25,47,50],"Pauli":[16],"directions":[17],"of":[18,37,62,68,110,144],"weights":[19],"two":[22],"$D^2$":[24],"all":[26],"higher":[27],"weights.":[28],"We":[29],"prove":[30],"that":[31],"distance":[33,154],"from":[34,107],"identity":[36],"a":[38,66,108],"Haar-random":[39],"element,":[40],"normalized":[41],"by":[42,114],"$D$,":[43],"converges":[44],"to":[45,100,130],"$\u03c0/\\sqrt3$":[46],"probability":[48],"$L^p$":[51],"for":[52,86],"every":[53],"$1\\le":[54],"p<\\infty$.":[55],"Quantitatively,":[56],"it":[57],"lies":[58],"within":[59],"$O(D^{-1/8}(\\log":[60],"D)^{1/2})$":[61],"this":[63],"limit":[64],"outside":[65],"set":[67],"Haar":[69],"measure":[70,82],"at":[71,83],"most":[72,84],"$\\exp\\{-\u03a9(D^{7/4}\\log":[73],"D)\\}$.":[74],"For":[75],"each":[76],"fixed":[77],"$0\u03c0/\\sqrt3$,":[78],"its":[79],"complement":[80],"has":[81],"$e^{-c_xD^2}$":[85],"some":[87],"$c_x>0$.":[88],"As":[89],"$x\\uparrow\u03c0/\\sqrt3$,":[90],"lower":[92],"upper":[94,104,155],"logarithmic":[95],"rates":[96],"are":[97],"both":[98],"asymptotic":[99],"$(\u03c0^2/3-x^2)^2/(16\u03b6(3))$.":[101],"The":[102],"small-ball":[103],"bound":[105],"follows":[106],"comparison":[109],"Jacobi":[111],"determinants,":[112],"obtained":[113],"rescaling":[115],"linearized":[117],"geodesic":[118],"equations":[119],"applying":[121],"Kato":[122],"transport.":[123],"Weyl":[124],"integration":[125],"reduces":[126],"remaining":[128],"integral":[129],"an":[131],"Abel-regularized":[132],"logarithmic-energy":[133],"estimate":[134],"on":[135],"circle.":[137],"A":[138],"centered":[139],"principal":[140],"logarithm":[141],"concentration":[143],"circular":[146],"unitary":[147],"ensemble":[148],"eigenangle":[149],"second":[150],"moment":[151],"give":[152],"bound.":[156]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-11T00:00:00"}
