{"id":"https://openalex.org/W7220957172","doi":"https://doi.org/10.48550/arxiv.2610.08701","title":"Large Growth Happens: Gaussian Elimination with Partial Pivoting on Random Matrices","display_name":"Large Growth Happens: Gaussian Elimination with Partial Pivoting on Random Matrices","publication_year":2026,"publication_date":"2026-10-06","ids":{"openalex":"https://openalex.org/W7220957172","doi":"https://doi.org/10.48550/arxiv.2610.08701"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.08701","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.08701","pdf_url":"https://arxiv.org/pdf/2610.08701","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/2610.08701","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5061644327","display_name":"Daniel Alan Spielman","orcid":"https://orcid.org/0000-0002-8958-879X","observed_orcids":["https://orcid.org/0000-0002-8958-879X"]},"institutions":[{"id":"https://openalex.org/I32971472","display_name":"Yale University","ror":"https://ror.org/03v76x132","country_code":"US","type":"education","lineage":["https://openalex.org/I32971472"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Spielman, Daniel A.","raw_affiliation_strings":["Department of Computer Science , Yale University","Department of Statistics and Data Science , Yale University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer Science , Yale University","institution_ids":["https://openalex.org/I32971472"]},{"raw_affiliation_string":"Department of Statistics and Data Science , Yale University","institution_ids":["https://openalex.org/I32971472"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5157892625","display_name":"Xifan Yu","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I32971472","display_name":"Yale University","ror":"https://ror.org/03v76x132","country_code":"US","type":"education","lineage":["https://openalex.org/I32971472"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Yu, Xifan","raw_affiliation_strings":["Department of Computer Science , Yale University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Computer Science , Yale University","institution_ids":["https://openalex.org/I32971472"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I32971472"],"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.9526000022888184,"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.9526000022888184,"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.015799999237060547,"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/T11697","display_name":"Numerical Methods and Algorithms","score":0.006800000090152025,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/gaussian-elimination","display_name":"Gaussian elimination","score":0.9020000100135803},{"id":"https://openalex.org/keywords/partial-pivoting","display_name":"partial pivoting","score":0.9990000128746033},{"id":"https://openalex.org/keywords/gaussian-random-matrices","display_name":"Gaussian random matrices","score":0.8690000176429749},{"id":"https://openalex.org/keywords/lower-bounds","display_name":"lower bounds","score":0.6460000276565552},{"id":"https://openalex.org/keywords/smoothed-analysis","display_name":"smoothed analysis","score":0.8050000071525574},{"id":"https://openalex.org/keywords/average-case-analysis","display_name":"average-case analysis","score":0.9210000038146973},{"id":"https://openalex.org/keywords/numerical-linear-algebra","display_name":"numerical linear algebra","score":0.6639999747276306}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7476000189781189},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.6658999919891357},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.6060000061988831},{"id":"https://openalex.org/C51267290","wikidata":"https://www.wikidata.org/wiki/Q5527848","display_name":"Gaussian random field","level":4,"score":0.5304999947547913},{"id":"https://openalex.org/C132878287","wikidata":"https://www.wikidata.org/wiki/Q1671727","display_name":"Inverse Gaussian distribution","level":3,"score":0.4625999927520752},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.4595000147819519},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4580000042915344},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.4537999927997589},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.43529999256134033},{"id":"https://openalex.org/C64812099","wikidata":"https://www.wikidata.org/wiki/Q176604","display_name":"Random matrix","level":3,"score":0.41449999809265137},{"id":"https://openalex.org/C126312332","wikidata":"https://www.wikidata.org/wiki/Q2658","display_name":"Gaussian elimination","level":3,"score":0.38420000672340393},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.3799999952316284},{"id":"https://openalex.org/C61326573","wikidata":"https://www.wikidata.org/wiki/Q1496376","display_name":"Gaussian process","level":3,"score":0.35749998688697815},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.30489999055862427},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3005000054836273},{"id":"https://openalex.org/C4199805","wikidata":"https://www.wikidata.org/wiki/Q2725903","display_name":"Gaussian noise","level":2,"score":0.2782000005245209},{"id":"https://openalex.org/C18912344","wikidata":"https://www.wikidata.org/wiki/Q866810","display_name":"Generalized inverse Gaussian distribution","level":5,"score":0.27630001306533813}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.08701","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.08701","pdf_url":"https://arxiv.org/pdf/2610.08701","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.2610.08701","doi":"https://doi.org/10.48550/arxiv.2610.08701","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.08701","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:2610.08701","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.08701","pdf_url":"https://arxiv.org/pdf/2610.08701","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":[{"id":"https://openalex.org/F4320337345","display_name":"Office of Naval Research","ror":"https://ror.org/00rk2pe57"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7220957172.pdf","grobid_xml":"https://content.openalex.org/works/W7220957172.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7220730786","https://openalex.org/W7220659047","https://openalex.org/W7203927659","https://openalex.org/W2170078634","https://openalex.org/W4395000858","https://openalex.org/W2133428703","https://openalex.org/W2108029014","https://openalex.org/W2115780241","https://openalex.org/W2141684379","https://openalex.org/W4205662659"],"abstract_inverted_index":{"We":[0],"prove":[1],"that":[2,5],"the":[3,37,61,70,77,83],"probability":[4,38,84],"Gaussian":[6,15,67,89],"elimination":[7],"with":[8],"partial":[9],"pivoting":[10],"on":[11,82],"$n":[12],"\\times":[13],"n$":[14],"random":[16,90],"matrices":[17],"has":[18],"growth":[19,40,87],"$\u03c1$":[20],"is":[21,45,76],"at":[22,46],"least":[23,47],"inverse":[24],"quasi-polynomial:":[25],"$\u03a9\\left(\\exp\\left(-c":[26],"\\log^2":[27],"\\left(\\frac\u03c1{\\sqrt{n}}\\right)\\log(n)\\right)\\right)$":[28],"for":[29,88],"some":[30],"constant":[31],"$c":[32],">":[33],"0$.":[34],"In":[35],"particular,":[36],"of":[39,66,72,85],"$\u03c1\\ge":[41],"n^{1/2":[42],"+":[43],"\u03b5}$":[44],"$\\exp(-":[48],"O":[49],"(\u03b5^2":[50],"\\log^3":[51],"(n)":[52],"))$.":[53],"This":[54],"lower":[55,80],"bound":[56,81],"breaks":[57],"standard":[58],"conjectures":[59],"in":[60],"smoothed":[62],"and":[63],"average-case":[64],"analysis":[65],"elimination.":[68],"To":[69],"best":[71],"our":[73],"knowledge,":[74],"it":[75],"first":[78],"non-trivial":[79],"large":[86],"matrices.":[91]},"counts_by_year":[],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2026-10-09T00:00:00"}