{"id":"https://openalex.org/W7220387602","doi":"https://doi.org/10.48550/arxiv.2610.03555","title":"Limit law for Brownian cover time of the two dimensional torus","display_name":"Limit law for Brownian cover time of the two dimensional torus","publication_year":2026,"publication_date":"2026-10-02","ids":{"openalex":"https://openalex.org/W7220387602","doi":"https://doi.org/10.48550/arxiv.2610.03555"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.03555","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.03555","pdf_url":"https://arxiv.org/pdf/2610.03555","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/2610.03555","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5019243217","display_name":"Amir Dembo","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Dembo, Amir","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5033076994","display_name":"Jay Rosen","orcid":"https://orcid.org/0000-0001-7372-725X","observed_orcids":["https://orcid.org/0000-0001-7372-725X"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Rosen, Jay","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5000594503","display_name":"Ofer Zeitouni","orcid":"https://orcid.org/0000-0002-2520-1525","observed_orcids":["https://orcid.org/0000-0002-2520-1525"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Zeitouni, Ofer","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/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.9779999852180481,"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":0.9779999852180481,"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/T13500","display_name":"Probability and Statistical Research","score":0.006500000134110451,"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/T11830","display_name":"Point processes and geometric inequalities","score":0.0026000000070780516,"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/two-dimensional-torus","display_name":"two-dimensional torus","score":0.902999997138977},{"id":"https://openalex.org/keywords/limit-laws","display_name":"limit laws","score":0.6869999766349792},{"id":"https://openalex.org/keywords/gumbel-distribution","display_name":"Gumbel distribution","score":0.921999990940094},{"id":"https://openalex.org/keywords/brownian-motion","display_name":"Brownian motion","score":0.656000018119812},{"id":"https://openalex.org/keywords/weak-convergence","display_name":"weak convergence","score":0.5590000152587891}],"concepts":[{"id":"https://openalex.org/C9767117","wikidata":"https://www.wikidata.org/wiki/Q12510","display_name":"Torus","level":2,"score":0.7461000084877014},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7233999967575073},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.6247000098228455},{"id":"https://openalex.org/C112401455","wikidata":"https://www.wikidata.org/wiki/Q178036","display_name":"Brownian motion","level":2,"score":0.593999981880188},{"id":"https://openalex.org/C2780428219","wikidata":"https://www.wikidata.org/wiki/Q16952335","display_name":"Cover (algebra)","level":2,"score":0.5138999819755554},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4986000061035156},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.45719999074935913},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.44040000438690186},{"id":"https://openalex.org/C137610916","wikidata":"https://www.wikidata.org/wiki/Q1096862","display_name":"Gumbel distribution","level":3,"score":0.41260001063346863},{"id":"https://openalex.org/C78651780","wikidata":"https://www.wikidata.org/wiki/Q628309","display_name":"Brownian bridge","level":3,"score":0.3628999888896942},{"id":"https://openalex.org/C122637931","wikidata":"https://www.wikidata.org/wiki/Q118084","display_name":"Unit (ring theory)","level":2,"score":0.33090001344680786},{"id":"https://openalex.org/C17825722","wikidata":"https://www.wikidata.org/wiki/Q17285","display_name":"Plane (geometry)","level":2,"score":0.32910001277923584},{"id":"https://openalex.org/C7356347","wikidata":"https://www.wikidata.org/wiki/Q1307838","display_name":"Unit square","level":2,"score":0.31279999017715454},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.31029999256134033},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.30469998717308044},{"id":"https://openalex.org/C75432250","wikidata":"https://www.wikidata.org/wiki/Q29654355","display_name":"Real line","level":2,"score":0.2799000144004822},{"id":"https://openalex.org/C104114177","wikidata":"https://www.wikidata.org/wiki/Q79782","display_name":"Motion (physics)","level":2,"score":0.2757999897003174},{"id":"https://openalex.org/C53920899","wikidata":"https://www.wikidata.org/wiki/Q4976524","display_name":"Brownian excursion","level":5,"score":0.27459999918937683},{"id":"https://openalex.org/C87040749","wikidata":"https://www.wikidata.org/wiki/Q428971","display_name":"Power law","level":2,"score":0.27320000529289246},{"id":"https://openalex.org/C88871306","wikidata":"https://www.wikidata.org/wiki/Q7208287","display_name":"Point process","level":2,"score":0.2727999985218048},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.26829999685287476},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.2653999924659729},{"id":"https://openalex.org/C108819105","wikidata":"https://www.wikidata.org/wiki/Q1143293","display_name":"Fractional Brownian motion","level":3,"score":0.25850000977516174},{"id":"https://openalex.org/C8272713","wikidata":"https://www.wikidata.org/wiki/Q176737","display_name":"Stochastic process","level":2,"score":0.257999986410141},{"id":"https://openalex.org/C126875415","wikidata":"https://www.wikidata.org/wiki/Q203425","display_name":"Unit circle","level":2,"score":0.2572000026702881},{"id":"https://openalex.org/C91229774","wikidata":"https://www.wikidata.org/wiki/Q193885","display_name":"Infinitesimal","level":2,"score":0.2549000084400177},{"id":"https://openalex.org/C24084028","wikidata":"https://www.wikidata.org/wiki/Q7429781","display_name":"Scaling limit","level":3,"score":0.2522999942302704},{"id":"https://openalex.org/C61797465","wikidata":"https://www.wikidata.org/wiki/Q1188986","display_name":"Term (time)","level":2,"score":0.2513999938964844},{"id":"https://openalex.org/C185767445","wikidata":"https://www.wikidata.org/wiki/Q207952","display_name":"Law of large numbers","level":3,"score":0.250900000333786},{"id":"https://openalex.org/C2780009758","wikidata":"https://www.wikidata.org/wiki/Q6804172","display_name":"Measure (data warehouse)","level":2,"score":0.250900000333786}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.03555","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.03555","pdf_url":"https://arxiv.org/pdf/2610.03555","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.2610.03555","doi":"https://doi.org/10.48550/arxiv.2610.03555","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.03555","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.03555","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.03555","pdf_url":"https://arxiv.org/pdf/2610.03555","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/G6077833213","display_name":"Random walks and graphs, Gibbs measures, dynamics and universality.","funder_award_id":"2348142","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7220387602.pdf","grobid_xml":"https://content.openalex.org/works/W7220387602.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W7131689299"],"related_works":["https://openalex.org/W7216021107","https://openalex.org/W4385963164","https://openalex.org/W2257118316","https://openalex.org/W2099322621","https://openalex.org/W2803793785","https://openalex.org/W7097101397","https://openalex.org/W1987243055","https://openalex.org/W2120076230","https://openalex.org/W2114344097","https://openalex.org/W2950100038"],"abstract_inverted_index":{"Let":[0],"$\\mathcal{C}_\\varepsilon$":[1],"denote":[2],"the":[3,19,36],"time":[4],"it":[5],"takes":[6],"a":[7,50],"Brownian":[8],"motion":[9],"to":[10,49],"come":[11],"within":[12],"distance":[13],"$\\varepsilon$":[14],"of":[15,24],"every":[16],"point":[17],"in":[18,43],"two":[20],"dimensional":[21],"torus":[22],"$\\mathbb{T}^2$":[23],"unit":[25],"area.":[26],"We":[27],"prove":[28],"that":[29],"for":[30],"an":[31],"explicit":[32],"non-random":[33],"centering":[34],"$m_\\varepsilon$,":[35],"variables":[37],"$\\{\\sqrt{\\mathcal{C}_{\\varepsilon}}":[38],"-":[39],"m_\\varepsilon":[40],"\\}$":[41],"converge":[42],"distribution":[44],"as":[45],"$\\varepsilon":[46],"\\to":[47],"0$,":[48],"randomly":[51],"shifted":[52],"Gumbel":[53],"law.":[54]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-06T00:00:00"}