{"id":"https://openalex.org/W7213369901","doi":"https://doi.org/10.48550/arxiv.2609.14945","title":"Long-range expanders: construction and cutoff","display_name":"Long-range expanders: construction and cutoff","publication_year":2026,"publication_date":"2026-09-14","ids":{"openalex":"https://openalex.org/W7213369901","doi":"https://doi.org/10.48550/arxiv.2609.14945"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.14945","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.14945","pdf_url":"https://arxiv.org/pdf/2609.14945","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/2609.14945","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5051371597","display_name":"Dylan J. Altschuler","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Altschuler, Dylan J.","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5051371597"],"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.7095000147819519,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.7095000147819519,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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":0.10100000351667404,"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/T10849","display_name":"Finite Group Theory Research","score":0.03889999911189079,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/ramanujan-graphs","display_name":"Ramanujan graphs","score":0.8650000095367432},{"id":"https://openalex.org/keywords/spectral-expansion","display_name":"spectral expansion","score":0.7310000061988831},{"id":"https://openalex.org/keywords/low-distortion-embeddings","display_name":"low-distortion embeddings","score":0.734000027179718},{"id":"https://openalex.org/keywords/random-walks-on-graphs","display_name":"random walks on graphs","score":0.6000000238418579},{"id":"https://openalex.org/keywords/cutoff-phenomenon","display_name":"cutoff phenomenon","score":0.8960000276565552},{"id":"https://openalex.org/keywords/mixing-rate","display_name":"mixing rate","score":0.6850000023841858},{"id":"https://openalex.org/keywords/girth","display_name":"girth","score":0.6480000019073486},{"id":"https://openalex.org/keywords/random-regular-graphs","display_name":"random regular graphs","score":0.6489999890327454}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8815000057220459},{"id":"https://openalex.org/C131220774","wikidata":"https://www.wikidata.org/wiki/Q1441456","display_name":"Ramanujan's sum","level":2,"score":0.7846999764442444},{"id":"https://openalex.org/C2778217198","wikidata":"https://www.wikidata.org/wiki/Q556977","display_name":"Cutoff","level":2,"score":0.7771999835968018},{"id":"https://openalex.org/C154547637","wikidata":"https://www.wikidata.org/wiki/Q776602","display_name":"Expander graph","level":3,"score":0.659600019454956},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.5598999857902527},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4431000053882599},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.43140000104904175},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.42590001225471497},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.4113999903202057},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.38119998574256897},{"id":"https://openalex.org/C188370112","wikidata":"https://www.wikidata.org/wiki/Q173091","display_name":"Transcendental number","level":2,"score":0.37279999256134033},{"id":"https://openalex.org/C97489613","wikidata":"https://www.wikidata.org/wiki/Q842755","display_name":"Peano axioms","level":2,"score":0.37049999833106995},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.3653999865055084},{"id":"https://openalex.org/C120204988","wikidata":"https://www.wikidata.org/wiki/Q859174","display_name":"Cayley graph","level":3,"score":0.36500000953674316},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.33149999380111694},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.33149999380111694},{"id":"https://openalex.org/C191399111","wikidata":"https://www.wikidata.org/wiki/Q64861","display_name":"Transitive relation","level":2,"score":0.31459999084472656},{"id":"https://openalex.org/C2776809875","wikidata":"https://www.wikidata.org/wiki/Q1375963","display_name":"Converse","level":2,"score":0.3140999972820282},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.30880001187324524},{"id":"https://openalex.org/C61326573","wikidata":"https://www.wikidata.org/wiki/Q1496376","display_name":"Gaussian process","level":3,"score":0.3068000078201294},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.29330000281333923},{"id":"https://openalex.org/C85307737","wikidata":"https://www.wikidata.org/wiki/Q751969","display_name":"Coset","level":2,"score":0.28940001130104065},{"id":"https://openalex.org/C178009071","wikidata":"https://www.wikidata.org/wiki/Q93344","display_name":"Trigonometric functions","level":2,"score":0.2879999876022339},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.2768999934196472},{"id":"https://openalex.org/C121194460","wikidata":"https://www.wikidata.org/wiki/Q856741","display_name":"Random walk","level":2,"score":0.26809999346733093},{"id":"https://openalex.org/C203024314","wikidata":"https://www.wikidata.org/wiki/Q1365258","display_name":"Fourier analysis","level":3,"score":0.2599000036716461},{"id":"https://openalex.org/C6142545","wikidata":"https://www.wikidata.org/wiki/Q1455881","display_name":"Cutoff frequency","level":2,"score":0.2533000111579895},{"id":"https://openalex.org/C199422724","wikidata":"https://www.wikidata.org/wiki/Q41118","display_name":"Quotient","level":2,"score":0.25}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.14945","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.14945","pdf_url":"https://arxiv.org/pdf/2609.14945","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.2609.14945","doi":"https://doi.org/10.48550/arxiv.2609.14945","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.14945","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:2609.14945","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.14945","pdf_url":"https://arxiv.org/pdf/2609.14945","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/W7213369901.pdf","grobid_xml":"https://content.openalex.org/works/W7213369901.grobid-xml"},"referenced_works_count":2,"referenced_works":["https://openalex.org/W4403963868","https://openalex.org/W6910377272"],"related_works":["https://openalex.org/W6889125436","https://openalex.org/W4376651343","https://openalex.org/W3037600800","https://openalex.org/W2117306309","https://openalex.org/W7169604873","https://openalex.org/W4297847074","https://openalex.org/W2443872466","https://openalex.org/W2141722918","https://openalex.org/W3165775096","https://openalex.org/W3044513950"],"abstract_inverted_index":{"Long-range":[0],"expansion":[1,97,114,143],"is":[2,119,155],"a":[3,134,158,189],"combinatorial":[4],"graph":[5],"property":[6,165],"introduced":[7],"to":[8,16],"construct":[9],"metric":[10,28],"spaces":[11],"with":[12,54,89],"strong":[13,64],"quantitative":[14],"obstructions":[15],"low-distortion":[17],"embeddings.":[18],"Such":[19],"extremal":[20],"constructions":[21,38,79],"are":[22,43,104,115,166],"motivated":[23],"by":[24,66,202],"fundamental":[25],"questions":[26],"in":[27,63],"geometry":[29],"and":[30,101,112,162,183,204],"nonlinear":[31],"functional":[32],"analysis.":[33],"However,":[34],"the":[35,47,69,109,139,145,163,171,179,206],"only":[36],"known":[37],"of":[39,49,142,148,209],"long-range":[40,73,86,96,107],"expander":[41,87],"sequences":[42,88],"random":[44,149,172],"regular":[45],"graphs;":[46],"problem":[48],"constructing":[50],"explicit":[51,85],"sequences,":[52],"especially":[53],"logarithmic":[55,90],"girth,":[56],"remains":[57],"open.":[58],"We":[59,92,129,168],"resolve":[60],"this":[61],"question":[62],"form":[65],"proving":[67],"that":[68,95,170],"Ramanujan":[70,78,110,164,190,200],"condition":[71],"implies":[72,98],"expansion.":[74],"In":[75],"particular,":[76],"classical":[77],"(such":[80],"as":[81,187],"LPS":[82],"graphs)":[83],"give":[84],"girth.":[91],"also":[93],"clarify":[94],"spectral":[99,113,213],"expansion,":[100,108],"both":[102],"implications":[103],"strict.":[105],"While":[106],"property,":[111],"strictly":[116],"separated,":[117],"it":[118],"\\textit{a":[120],"priori}":[121],"not":[122],"clear":[123],"how":[124,137],"robust":[125],"these":[126],"gaps":[127],"are.":[128],"next":[130],"compare":[131],"them":[132],"from":[133,157],"dynamics":[135,159],"perspective:":[136],"do":[138],"different":[140],"notions":[141],"affect":[144],"mixing":[146],"rate":[147],"walks?":[150],"Our":[151],"second":[152],"main":[153],"result":[154],"that,":[156],"perspective,":[160],"LRE":[161,176],"equivalent.":[167],"show":[169],"walk":[173],"on":[174,188],"any":[175],"sequence":[177],"has":[178],"same":[180],"cutoff":[181,198,210],"location":[182],"Gaussian":[184],"limit":[185],"profile":[186],"graph.":[191],"This":[192],"incidentally":[193],"offers":[194],"intermediate":[195],"progress":[196],"between":[197],"for":[199,211],"graphs---proven":[201],"Lubetzky":[203],"Peres---and":[205],"long-standing":[207],"conjecture":[208],"transitive":[212],"expanders.":[214]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-11-09T13:22:03.289127","created_date":"2026-09-16T00:00:00"}
