{"id":"https://openalex.org/W7214752091","doi":"https://doi.org/10.48550/arxiv.2609.35213","title":"Very sharp distance and range transitions for random walk bridges on Ramanujan graphs","display_name":"Very sharp distance and range transitions for random walk bridges on Ramanujan graphs","publication_year":2026,"publication_date":"2026-09-28","ids":{"openalex":"https://openalex.org/W7214752091","doi":"https://doi.org/10.48550/arxiv.2609.35213"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.35213","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.35213","pdf_url":"https://arxiv.org/pdf/2609.35213","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","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.35213","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5059311017","display_name":"Ita\u00ef Benjamini","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Benjamini, Itai","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5059311017"],"corresponding_institution_ids":[],"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/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.9320999979972839,"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":0.9320999979972839,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.040800001472234726,"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":0.00430000014603138,"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/ramanujan-graphs","display_name":"Ramanujan graphs","score":0.9950000047683716},{"id":"https://openalex.org/keywords/vertex-transitive-graphs","display_name":"vertex-transitive graphs","score":0.9459999799728394},{"id":"https://openalex.org/keywords/critical-window","display_name":"critical window","score":0.5870000123977661},{"id":"https://openalex.org/keywords/regular-trees","display_name":"regular trees","score":0.6140000224113464},{"id":"https://openalex.org/keywords/limiting-laws","display_name":"limiting laws","score":0.6890000104904175},{"id":"https://openalex.org/keywords/random-walks-on-graphs","display_name":"random walks on graphs","score":0.6050000190734863}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7850000262260437},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.6707000136375427},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6679999828338623},{"id":"https://openalex.org/C121194460","wikidata":"https://www.wikidata.org/wiki/Q856741","display_name":"Random walk","level":2,"score":0.6668999791145325},{"id":"https://openalex.org/C39927690","wikidata":"https://www.wikidata.org/wiki/Q11197","display_name":"Logarithm","level":2,"score":0.6051999926567078},{"id":"https://openalex.org/C131220774","wikidata":"https://www.wikidata.org/wiki/Q1441456","display_name":"Ramanujan's sum","level":2,"score":0.5044999718666077},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.4932999908924103},{"id":"https://openalex.org/C204323151","wikidata":"https://www.wikidata.org/wiki/Q905424","display_name":"Range (aeronautics)","level":2,"score":0.48809999227523804},{"id":"https://openalex.org/C188198153","wikidata":"https://www.wikidata.org/wiki/Q1613840","display_name":"Limiting","level":2,"score":0.4830999970436096},{"id":"https://openalex.org/C113174947","wikidata":"https://www.wikidata.org/wiki/Q2859736","display_name":"Tree (set theory)","level":2,"score":0.4271000027656555},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4226999878883362},{"id":"https://openalex.org/C47458327","wikidata":"https://www.wikidata.org/wiki/Q910404","display_name":"Random graph","level":3,"score":0.3962000012397766},{"id":"https://openalex.org/C154547637","wikidata":"https://www.wikidata.org/wiki/Q776602","display_name":"Expander graph","level":3,"score":0.36550000309944153},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.3555999994277954},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.35429999232292175},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.3538999855518341},{"id":"https://openalex.org/C100776233","wikidata":"https://www.wikidata.org/wiki/Q2532492","display_name":"Bridge (graph theory)","level":2,"score":0.2964000105857849},{"id":"https://openalex.org/C155155215","wikidata":"https://www.wikidata.org/wiki/Q7448025","display_name":"Self-avoiding walk","level":3,"score":0.2777000069618225},{"id":"https://openalex.org/C17825722","wikidata":"https://www.wikidata.org/wiki/Q17285","display_name":"Plane (geometry)","level":2,"score":0.27639999985694885},{"id":"https://openalex.org/C20353970","wikidata":"https://www.wikidata.org/wiki/Q1056998","display_name":"Simple random sample","level":3,"score":0.266400009393692},{"id":"https://openalex.org/C24723263","wikidata":"https://www.wikidata.org/wiki/Q1571884","display_name":"Critical distance","level":4,"score":0.26100000739097595},{"id":"https://openalex.org/C90377204","wikidata":"https://www.wikidata.org/wiki/Q1052594","display_name":"Uniform boundedness","level":3,"score":0.25850000977516174},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.25130000710487366}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.35213","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.35213","pdf_url":"https://arxiv.org/pdf/2609.35213","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.2609.35213","doi":"https://doi.org/10.48550/arxiv.2609.35213","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.35213","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":"public-domain","license_id":"https://openalex.org/licenses/public-domain","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.35213","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.35213","pdf_url":"https://arxiv.org/pdf/2609.35213","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/W7214752091.pdf","grobid_xml":"https://content.openalex.org/works/W7214752091.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W900403624","https://openalex.org/W7098193893","https://openalex.org/W2064561281","https://openalex.org/W2620687821","https://openalex.org/W2212192431","https://openalex.org/W2104099158","https://openalex.org/W3098243456","https://openalex.org/W2940079884","https://openalex.org/W3147437636","https://openalex.org/W2046857250"],"abstract_inverted_index":{"For":[0],"vertex-transitive":[1],"Ramanujan":[2],"graphs":[3],"with":[4],"logarithmic":[5],"girth,":[6],"a":[7,27,41,90],"simple":[8],"random":[9],"walk":[10],"bridge":[11],"of":[12,14,23,72],"length":[13],"order":[15,33,37],"$\\log":[16,38],"N$,":[17],"where":[18],"$N$":[19],"is":[20],"the":[21,24,54,70,77,82,86,100],"size":[22],"graph,":[25],"has":[26,89],"maximum":[28],"distance":[29],"that":[30],"changes":[31],"from":[32,58],"$\\sqrt{\\log":[34],"N}$":[35],"to":[36,53],"N$":[39],"in":[40],"bounded":[42],"critical":[43,79],"window.":[44,101],"We":[45],"prove":[46],"this":[47],"by":[48],"separating":[49],"bridges":[50],"whose":[51,60,94],"lifts":[52,61],"regular":[55],"tree":[56],"close":[57],"those":[59],"do":[62],"not.":[63],"A":[64],"uniform":[65],"two-term":[66],"return":[67],"estimate":[68],"determines":[69],"probabilities":[71],"these":[73],"two":[74],"cases":[75],"and":[76],"real-valued":[78],"center.":[80],"In":[81],"same":[83],"$O(1)$":[84],"window,":[85],"normalized":[87],"range":[88],"two-point":[91],"limiting":[92],"law":[93],"mixture":[95],"weights":[96],"vary":[97],"nontrivially":[98],"across":[99]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-09-30T00:00:00"}
