{"id":"https://openalex.org/W7213602256","doi":"https://doi.org/10.48550/arxiv.2609.19474","title":"A Universal Law of Large Numbers for Extreme Cycles in Random \u010cech Complexes","display_name":"A Universal Law of Large Numbers for Extreme Cycles in Random \u010cech Complexes","publication_year":2026,"publication_date":"2026-09-16","ids":{"openalex":"https://openalex.org/W7213602256","doi":"https://doi.org/10.48550/arxiv.2609.19474"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.19474","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.19474","pdf_url":"https://arxiv.org/pdf/2609.19474","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/2609.19474","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5070671339","display_name":"Omer Bobrowski","orcid":"https://orcid.org/0000-0002-0860-7099","observed_orcids":["https://orcid.org/0000-0002-0860-7099"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Bobrowski, Omer","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5049296186","display_name":"Primo\u017e \u0160kraba","orcid":"https://orcid.org/0000-0002-5300-8984","observed_orcids":["https://orcid.org/0000-0002-5300-8984"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Skraba, Primoz","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"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/T12536","display_name":"Topological and Geometric Data Analysis","score":0.9932000041007996,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12536","display_name":"Topological and Geometric Data Analysis","score":0.9932000041007996,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11830","display_name":"Point processes and geometric inequalities","score":0.0035000001080334187,"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/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.0003000000142492354,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/k-cycles","display_name":"k-cycles","score":0.781000018119812},{"id":"https://openalex.org/keywords/law-of-large-numbers","display_name":"law of large numbers","score":0.9010000228881836},{"id":"https://openalex.org/keywords/universality","display_name":"universality","score":0.8980000019073486},{"id":"https://openalex.org/keywords/poisson-point-process","display_name":"Poisson point process","score":0.6790000200271606},{"id":"https://openalex.org/keywords/covering-density","display_name":"covering density","score":0.6100000143051147},{"id":"https://openalex.org/keywords/isoperimetric-inequality","display_name":"isoperimetric inequality","score":0.9639999866485596},{"id":"https://openalex.org/keywords/simplicial-complex","display_name":"simplicial complexes","score":0.7490000128746033},{"id":"https://openalex.org/keywords/topological-data-analysis","display_name":"topological data analysis","score":0.7390000224113464},{"id":"https://openalex.org/keywords/sphere-covering","display_name":"sphere covering","score":0.45500001311302185}],"concepts":[{"id":"https://openalex.org/C140142295","wikidata":"https://www.wikidata.org/wiki/Q617417","display_name":"Isoperimetric inequality","level":2,"score":0.8751999735832214},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8425999879837036},{"id":"https://openalex.org/C42747912","wikidata":"https://www.wikidata.org/wiki/Q1048447","display_name":"Multiplicative function","level":2,"score":0.671999990940094},{"id":"https://openalex.org/C185767445","wikidata":"https://www.wikidata.org/wiki/Q207952","display_name":"Law of large numbers","level":3,"score":0.5356000065803528},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.5325999855995178},{"id":"https://openalex.org/C75438885","wikidata":"https://www.wikidata.org/wiki/Q3403615","display_name":"Large deviations theory","level":2,"score":0.48820000886917114},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.4830999970436096},{"id":"https://openalex.org/C147581598","wikidata":"https://www.wikidata.org/wiki/Q729429","display_name":"Extreme value theory","level":2,"score":0.45509999990463257},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4544999897480011},{"id":"https://openalex.org/C188198153","wikidata":"https://www.wikidata.org/wiki/Q1613840","display_name":"Limiting","level":2,"score":0.4198000133037567},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.41370001435279846},{"id":"https://openalex.org/C18648836","wikidata":"https://www.wikidata.org/wiki/Q381892","display_name":"Compact space","level":2,"score":0.41119998693466187},{"id":"https://openalex.org/C183992945","wikidata":"https://www.wikidata.org/wiki/Q2495574","display_name":"Universality (dynamical systems)","level":2,"score":0.40880000591278076},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.37950000166893005},{"id":"https://openalex.org/C149441793","wikidata":"https://www.wikidata.org/wiki/Q200726","display_name":"Probability distribution","level":2,"score":0.3601999878883362},{"id":"https://openalex.org/C141042865","wikidata":"https://www.wikidata.org/wiki/Q200125","display_name":"Expected value","level":2,"score":0.35499998927116394},{"id":"https://openalex.org/C21031990","wikidata":"https://www.wikidata.org/wiki/Q355020","display_name":"Probability measure","level":2,"score":0.3357999920845032},{"id":"https://openalex.org/C84498859","wikidata":"https://www.wikidata.org/wiki/Q5158327","display_name":"Concentration of measure","level":2,"score":0.33180001378059387},{"id":"https://openalex.org/C65778772","wikidata":"https://www.wikidata.org/wiki/Q12345341","display_name":"Asymptotic distribution","level":3,"score":0.3043999969959259},{"id":"https://openalex.org/C122203268","wikidata":"https://www.wikidata.org/wiki/Q5862903","display_name":"Probability theory","level":2,"score":0.30379998683929443},{"id":"https://openalex.org/C165464430","wikidata":"https://www.wikidata.org/wiki/Q1570441","display_name":"Parameterized complexity","level":2,"score":0.30250000953674316},{"id":"https://openalex.org/C39947850","wikidata":"https://www.wikidata.org/wiki/Q729523","display_name":"Geometric distribution","level":3,"score":0.29910001158714294},{"id":"https://openalex.org/C197055811","wikidata":"https://www.wikidata.org/wiki/Q207522","display_name":"Probability density function","level":2,"score":0.29170000553131104},{"id":"https://openalex.org/C7605037","wikidata":"https://www.wikidata.org/wiki/Q327164","display_name":"Geometric probability","level":2,"score":0.29010000824928284},{"id":"https://openalex.org/C141547133","wikidata":"https://www.wikidata.org/wiki/Q7291996","display_name":"Random variate","level":3,"score":0.2897999882698059},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.28519999980926514},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.27489998936653137},{"id":"https://openalex.org/C8272713","wikidata":"https://www.wikidata.org/wiki/Q176737","display_name":"Stochastic process","level":2,"score":0.27129998803138733},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.2624000012874603},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.259799987077713},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.2567000091075897},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.2551000118255615}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.19474","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.19474","pdf_url":"https://arxiv.org/pdf/2609.19474","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.2609.19474","doi":"https://doi.org/10.48550/arxiv.2609.19474","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.19474","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:2609.19474","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.19474","pdf_url":"https://arxiv.org/pdf/2609.19474","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":[{"id":"https://openalex.org/G1677187438","display_name":null,"funder_award_id":"EP/Y008642/1","funder_id":"https://openalex.org/F4320319993","funder_display_name":"Leverhulme Trust"},{"id":"https://openalex.org/G3340358590","display_name":null,"funder_award_id":"RPG-2023-144","funder_id":"https://openalex.org/F4320319993","funder_display_name":"Leverhulme Trust"},{"id":"https://openalex.org/G395856516","display_name":"Topological Phase Transitions In Stochastic Geometry","funder_award_id":"EP/Y008642/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G6834831930","display_name":"Mathematical Foundations of Intelligence: An "Erlangen Programme" for AI","funder_award_id":"EP/Y028872/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G731302573","display_name":null,"funder_award_id":"EP/Y028872/1","funder_id":"https://openalex.org/F4320319993","funder_display_name":"Leverhulme Trust"},{"id":"https://openalex.org/G8592757369","display_name":null,"funder_award_id":"RPG-2023-144","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"}],"funders":[{"id":"https://openalex.org/F4320319993","display_name":"Leverhulme Trust","ror":"https://ror.org/012mzw131"},{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7213602256.pdf","grobid_xml":"https://content.openalex.org/works/W7213602256.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W7171939021"],"related_works":["https://openalex.org/W4383720948","https://openalex.org/W7221039829","https://openalex.org/W4242549976","https://openalex.org/W2009919500","https://openalex.org/W7155045645","https://openalex.org/W2168826068","https://openalex.org/W7221098025","https://openalex.org/W1965482388","https://openalex.org/W3134800435","https://openalex.org/W2119740160"],"abstract_inverted_index":{"We":[0,117],"study":[1],"the":[2,32,45,49,90,102,107,120,126,131,137,152,202],"maximal":[3,141],"multiplicative":[4,111],"persistence":[5,112],"of":[6,76,83,110,130,154,181,204],"$k$-cycles":[7,39],"in":[8,48,55],"random":[9],"\u010cech":[10],"complexes.":[11],"Let":[12],"$f:\\mathbb{R}^d\\to\\mathbb{R}$":[13],"be":[14,21],"a":[15,22,74,80,114,143,159,182,188,205],"probability":[16,103],"density":[17,104,129],"function,":[18],"let":[19,29],"${P}_n$":[20],"Poisson":[23],"process":[24],"with":[25,167],"intensity":[26],"$nf$,":[27],"and":[28,69,97,99],"$\u03a0_{k,n}$":[30],"denote":[31],"largest":[33],"death-to-birth":[34],"ratio":[35],"among":[36],"all":[37],"non-essential":[38],"($1\\le":[40],"k":[41],"\\le":[42],"d-1$).":[43],"For":[44],"uniform":[46],"distribution":[47],"unit":[50],"hypercube,":[51],"it":[52],"was":[53],"proved":[54],"[9]":[56],"that":[57,89,119,172,193],"$\u03a0_{k,n}":[58],"=":[59],"\u0398\\left(":[60],"\\left(\\frac{\\log":[61],"n}{\\log\\log":[62],"n}\\right)^{1/k}\\right)$.":[63],"In":[64],"this":[65,71,173],"paper":[66],"we":[67,87,170],"sharpen":[68],"extend":[70],"result":[72],"to":[73,157,196,201],"law":[75],"large":[77],"numbers,":[78],"for":[79],"broad":[81],"class":[82],"distributions.":[84],"Most":[85],"significantly,":[86],"show":[88,118,171],"limiting":[91,121,206],"constant":[92,122],"depends":[93],"only":[94],"on":[95,101,151],"$d$":[96],"$k$,":[98],"not":[100],"$f$.":[105],"Thus":[106],"extreme":[108],"value":[109],"exhibits":[113],"universality":[115],"phenomenon.":[116],"is":[123,175,187],"determined":[124],"by":[125,178],"asymptotic":[127],"covering":[128,199],"$k$-dimensional":[132],"sphere.":[133],"Our":[134],"proof":[135],"identifies":[136],"geometric":[138,189],"mechanism":[139],"underlying":[140],"cycles,":[142],"persistent":[144,161],"isoperimetric":[145,168],"inequality,":[146],"which":[147],"gives":[148],"sharp":[149],"bounds":[150],"number":[153],"points":[155],"needed":[156],"generate":[158],"highly":[160],"cycle.":[162,207],"By":[163],"combining":[164],"covering-density":[165],"estimates":[166],"inequalities,":[169],"minimum":[174],"asymptotically":[176],"attained":[177],"efficient":[179],"coverings":[180],"$k$-sphere.":[183],"A":[184],"key":[185],"ingredient":[186],"measure":[190],"theory":[191],"argument":[192],"uses":[194],"compactness":[195],"relate":[197],"discrete":[198],"counts":[200],"volume":[203]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-19T00:00:00"}