{"id":"https://openalex.org/W7203858610","doi":"https://doi.org/10.48550/arxiv.2608.18293","title":"Critical tensor covariance at the Marchenko-Pastur threshold","display_name":"Critical tensor covariance at the Marchenko-Pastur threshold","publication_year":2026,"publication_date":"2026-08-18","ids":{"openalex":"https://openalex.org/W7203858610","doi":"https://doi.org/10.48550/arxiv.2608.18293"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.18293","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.18293","pdf_url":"https://arxiv.org/pdf/2608.18293","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/2608.18293","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5084618257","display_name":"Xiaohui S. Xie","orcid":"https://orcid.org/0000-0002-5479-6345","observed_orcids":["https://orcid.org/0000-0002-5479-6345","https://orcid.org/0000-0001-9413-4461","https://orcid.org/0000-0002-2879-6464"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Xie, Xiaohui","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5084618257"],"corresponding_institution_ids":[],"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/T11716","display_name":"Random Matrices and Applications","score":0.9994999766349792,"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":0.9994999766349792,"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/T11022","display_name":"Spectral Theory in Mathematical Physics","score":0.0,"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/T11871","display_name":"Advanced Statistical Methods and Models","score":0.0,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/marchenko-pastur-law","display_name":"Marchenko-Pastur law","score":0.8460000157356262},{"id":"https://openalex.org/keywords/sample-covariance-matrix","display_name":"sample covariance matrix","score":0.6359999775886536},{"id":"https://openalex.org/keywords/lognormal-distribution","display_name":"lognormal distribution","score":0.6039999723434448},{"id":"https://openalex.org/keywords/critical-scale","display_name":"critical scale","score":0.4740000069141388},{"id":"https://openalex.org/keywords/free-probability","display_name":"free probability","score":0.4860000014305115},{"id":"https://openalex.org/keywords/random-matrices","display_name":"random matrices","score":0.36500000953674316}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8450000286102295},{"id":"https://openalex.org/C178650346","wikidata":"https://www.wikidata.org/wiki/Q201984","display_name":"Covariance","level":2,"score":0.6983000040054321},{"id":"https://openalex.org/C155281189","wikidata":"https://www.wikidata.org/wiki/Q3518150","display_name":"Tensor (intrinsic definition)","level":2,"score":0.5889999866485596},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5541999936103821},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.48890000581741333},{"id":"https://openalex.org/C166785042","wikidata":"https://www.wikidata.org/wiki/Q190391","display_name":"Central limit theorem","level":2,"score":0.4740000069141388},{"id":"https://openalex.org/C140532419","wikidata":"https://www.wikidata.org/wiki/Q249748","display_name":"Spectral radius","level":3,"score":0.47200000286102295},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.40939998626708984},{"id":"https://openalex.org/C179254644","wikidata":"https://www.wikidata.org/wiki/Q13222844","display_name":"Moment (physics)","level":2,"score":0.4032000005245209},{"id":"https://openalex.org/C95763700","wikidata":"https://www.wikidata.org/wiki/Q578985","display_name":"Convergence of random variables","level":3,"score":0.39640000462532043},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.39419999718666077},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.3935000002384186},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.3750999867916107},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.3555999994277954},{"id":"https://openalex.org/C204323151","wikidata":"https://www.wikidata.org/wiki/Q905424","display_name":"Range (aeronautics)","level":2,"score":0.3334999978542328},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.32839998602867126},{"id":"https://openalex.org/C98763669","wikidata":"https://www.wikidata.org/wiki/Q176645","display_name":"Markov chain","level":2,"score":0.31049999594688416},{"id":"https://openalex.org/C103692563","wikidata":"https://www.wikidata.org/wiki/Q5178900","display_name":"Covariance operator","level":3,"score":0.3059999942779541},{"id":"https://openalex.org/C185142706","wikidata":"https://www.wikidata.org/wiki/Q1134404","display_name":"Covariance matrix","level":2,"score":0.3010999858379364},{"id":"https://openalex.org/C57869625","wikidata":"https://www.wikidata.org/wiki/Q1783502","display_name":"Rate of convergence","level":3,"score":0.2953000068664551},{"id":"https://openalex.org/C178635117","wikidata":"https://www.wikidata.org/wiki/Q747499","display_name":"RADIUS","level":2,"score":0.29409998655319214},{"id":"https://openalex.org/C94360122","wikidata":"https://www.wikidata.org/wiki/Q1967557","display_name":"Spectral power distribution","level":2,"score":0.28540000319480896},{"id":"https://openalex.org/C117647685","wikidata":"https://www.wikidata.org/wiki/Q4241349","display_name":"Critical radius","level":3,"score":0.2732999920845032},{"id":"https://openalex.org/C62100291","wikidata":"https://www.wikidata.org/wiki/Q1936288","display_name":"Total variation","level":2,"score":0.2702000141143799},{"id":"https://openalex.org/C57945734","wikidata":"https://www.wikidata.org/wiki/Q7977941","display_name":"Weak convergence","level":3,"score":0.26330000162124634},{"id":"https://openalex.org/C155568369","wikidata":"https://www.wikidata.org/wiki/Q1428097","display_name":"Radius of convergence","level":3,"score":0.2612000107765198},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.25859999656677246},{"id":"https://openalex.org/C14928880","wikidata":"https://www.wikidata.org/wiki/Q1512559","display_name":"Normal convergence","level":4,"score":0.2542000114917755},{"id":"https://openalex.org/C185767445","wikidata":"https://www.wikidata.org/wiki/Q207952","display_name":"Law of large numbers","level":3,"score":0.2506999969482422}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.18293","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.18293","pdf_url":"https://arxiv.org/pdf/2608.18293","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.2608.18293","doi":"https://doi.org/10.48550/arxiv.2608.18293","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.18293","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:2608.18293","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.18293","pdf_url":"https://arxiv.org/pdf/2608.18293","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/W7203858610.pdf","grobid_xml":"https://content.openalex.org/works/W7203858610.grobid-xml"},"referenced_works_count":2,"referenced_works":["https://openalex.org/W1911497176","https://openalex.org/W7127417440"],"related_works":["https://openalex.org/W3213476537","https://openalex.org/W1911497176","https://openalex.org/W7171419512","https://openalex.org/W1991127564","https://openalex.org/W1979103890","https://openalex.org/W2162250452","https://openalex.org/W2741202445","https://openalex.org/W4401308047","https://openalex.org/W4308445590","https://openalex.org/W2066144838"],"abstract_inverted_index":{"For":[0],"a":[1],"centered,":[2],"variance-one":[3],"random":[4],"variable":[5,44],"$X$":[6],"with":[7,88],"finite":[8],"fourth":[9],"moment,":[10],"let":[11],"$x$":[12,39,77],"be":[13,129],"the":[14,27,34,42,69,84,108,111],"vector":[15],"of":[16,24,38,72,76,110],"square-free":[17],"degree-$d$":[18],"monomials":[19],"in":[20],"$n$":[21],"independent":[22,74],"copies":[23,75],"$X$.":[25],"At":[26],"critical":[28],"scale":[29],"$d^2/n":[30],"\\to":[31,65],"\u03bb\\in":[32],"(0,\\infty)$,":[33,68],"normalized":[35],"squared":[36],"length":[37,113,124],"converges":[40],"to":[41,97],"lognormal":[43],"$R":[45],"=":[46,53,101],"\\exp(\\sqrt{\u03bbv}\\,":[47],"Z":[48],"-":[49,56],"\u03bbv/2)$,":[50],"where":[51],"$v":[52],"\\mathbb{E}":[54],"X^4":[55],"1$":[57],"and":[58,91,125],"$Z$":[59],"is":[60],"standard":[61],"normal.":[62],"If":[63],"$\\binom{n}{d}/N":[64],"c":[66],"\\in":[67],"sample":[70,112],"covariance":[71],"$N$":[73],"has":[78],"an":[79],"almost-sure":[80],"limiting":[81],"spectral":[82],"law:":[83],"free":[85],"compound-Poisson":[86],"law":[87],"rate":[89],"$1/c$":[90],"jump":[92],"distribution":[93],"$\\operatorname{Law}(cR)$.":[94],"It":[95],"reduces":[96],"Marchenko-Pastur":[98],"when":[99,119],"$\u03bbv":[100],"0$.":[102],"The":[103],"proof":[104],"shows":[105],"that":[106],"subtracting":[107],"contribution":[109],"leaves":[114],"vanishing":[115],"quadratic-form":[116],"fluctuations,":[117],"even":[118],"$\\mathbb{E}":[120],"X^3":[121],"\\neq":[122],"0$;":[123],"direction":[126],"need":[127],"not":[128],"independent.":[130]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-08-22T00:00:00"}