{"id":"https://openalex.org/W7220987380","doi":"https://doi.org/10.48550/arxiv.2610.08517","title":"Sharp detection thresholds for random geometric graphs with polynomial average degrees","display_name":"Sharp detection thresholds for random geometric graphs with polynomial average degrees","publication_year":2026,"publication_date":"2026-10-06","ids":{"openalex":"https://openalex.org/W7220987380","doi":"https://doi.org/10.48550/arxiv.2610.08517"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.08517","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.08517","pdf_url":"https://arxiv.org/pdf/2610.08517","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/2610.08517","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5157972521","display_name":"Hang Du","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Du, Hang","raw_affiliation_strings":["Department of Mathematics , Massachusetts Institute of Technology , Cambridge , Massachusetts , USA,"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Massachusetts Institute of Technology , Cambridge , Massachusetts , USA,","institution_ids":["https://openalex.org/I63966007"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5059980139","display_name":"Cheng Mao","orcid":"https://orcid.org/0000-0003-3919-5637","observed_orcids":["https://orcid.org/0000-0003-3919-5637"]},"institutions":[{"id":"https://openalex.org/I170897317","display_name":"Duke University","ror":"https://ror.org/00py81415","country_code":"US","type":"education","lineage":["https://openalex.org/I170897317"]},{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Mao, Cheng","raw_affiliation_strings":["Department of Mathematics , Massachusetts Institute of Technology , Cambridge , Massachusetts , USA,","Fuqua School of Business , Duke University , Durham , North Carolina , USA,"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Massachusetts Institute of Technology , Cambridge , Massachusetts , USA,","institution_ids":["https://openalex.org/I63966007"]},{"raw_affiliation_string":"Fuqua School of Business , Duke University , Durham , North Carolina , USA,","institution_ids":["https://openalex.org/I170897317"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5069829605","display_name":"Nike Sun","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Sun, Nike","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5159902950","display_name":"Yihong Wu","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Wu, Yihong","raw_affiliation_strings":["Department of Mathematics , Massachusetts Institute of Technology , Cambridge , Massachusetts , USA,"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Massachusetts Institute of Technology , Cambridge , Massachusetts , USA,","institution_ids":["https://openalex.org/I63966007"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5157953304","display_name":"Jiaming Xu","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Xu, Jiaming","raw_affiliation_strings":["Department of Mathematics , Massachusetts Institute of Technology , Cambridge , Massachusetts , USA,"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics , Massachusetts Institute of Technology , Cambridge , Massachusetts , USA,","institution_ids":["https://openalex.org/I63966007"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":2,"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":0.6987000107765198,"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":0.6987000107765198,"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/T10064","display_name":"Complex Network Analysis Techniques","score":0.11079999804496765,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11716","display_name":"Random Matrices and Applications","score":0.039000000804662704,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/random-geometric-graphs","display_name":"random geometric graphs","score":0.9959999918937683},{"id":"https://openalex.org/keywords/erdos-renyi-graphs","display_name":"Erd\u0151s-R\u00e9nyi random graphs","score":0.7760000228881836},{"id":"https://openalex.org/keywords/kullback-leibler-divergence","display_name":"Kullback-Leibler divergence","score":0.9229999780654907},{"id":"https://openalex.org/keywords/indistinguishability","display_name":"indistinguishability","score":0.7540000081062317},{"id":"https://openalex.org/keywords/sparse-graphs","display_name":"sparse graphs","score":0.6830000281333923},{"id":"https://openalex.org/keywords/posterior-analysis","display_name":"posterior analysis","score":0.7630000114440918},{"id":"https://openalex.org/keywords/fourier-series","display_name":"Fourier series","score":0.6990000009536743},{"id":"https://openalex.org/keywords/spherical-harmonics-expansion","display_name":"spherical harmonic expansion","score":0.9480000138282776},{"id":"https://openalex.org/keywords/high-dimensional-spheres","display_name":"high-dimensional spheres","score":0.3790000081062317}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8422999978065491},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5296000242233276},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.4560999870300293},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.44269999861717224},{"id":"https://openalex.org/C179254644","wikidata":"https://www.wikidata.org/wiki/Q13222844","display_name":"Moment (physics)","level":2,"score":0.42080000042915344},{"id":"https://openalex.org/C102519508","wikidata":"https://www.wikidata.org/wiki/Q6520159","display_name":"Fourier transform","level":2,"score":0.4074000120162964},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.3797999918460846},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3693999946117401},{"id":"https://openalex.org/C207864730","wikidata":"https://www.wikidata.org/wiki/Q179467","display_name":"Fourier series","level":2,"score":0.3546999990940094},{"id":"https://openalex.org/C111030470","wikidata":"https://www.wikidata.org/wiki/Q1430460","display_name":"Curse of dimensionality","level":2,"score":0.34549999237060547},{"id":"https://openalex.org/C106195933","wikidata":"https://www.wikidata.org/wiki/Q7847935","display_name":"Truncation (statistics)","level":2,"score":0.3402999937534332},{"id":"https://openalex.org/C627467","wikidata":"https://www.wikidata.org/wiki/Q599027","display_name":"Harmonic function","level":2,"score":0.33390000462532043},{"id":"https://openalex.org/C203024314","wikidata":"https://www.wikidata.org/wiki/Q1365258","display_name":"Fourier analysis","level":3,"score":0.33009999990463257},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.32120001316070557},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.3172999918460846},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.31520000100135803},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3084999918937683},{"id":"https://openalex.org/C30609935","wikidata":"https://www.wikidata.org/wiki/Q7291969","display_name":"Random geometric graph","level":5,"score":0.3018999993801117},{"id":"https://openalex.org/C47458327","wikidata":"https://www.wikidata.org/wiki/Q910404","display_name":"Random graph","level":3,"score":0.30169999599456787},{"id":"https://openalex.org/C2777634741","wikidata":"https://www.wikidata.org/wiki/Q768993","display_name":"Wasserstein metric","level":2,"score":0.3003999888896942},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.2930999994277954},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.29120001196861267},{"id":"https://openalex.org/C207390915","wikidata":"https://www.wikidata.org/wiki/Q1230525","display_name":"Divergence (linguistics)","level":2,"score":0.28949999809265137},{"id":"https://openalex.org/C127934551","wikidata":"https://www.wikidata.org/wiki/Q1148098","display_name":"Harmonic","level":2,"score":0.28839999437332153},{"id":"https://openalex.org/C7605037","wikidata":"https://www.wikidata.org/wiki/Q327164","display_name":"Geometric probability","level":2,"score":0.2827000021934509},{"id":"https://openalex.org/C169391604","wikidata":"https://www.wikidata.org/wiki/Q7604402","display_name":"Statistical manifold","level":5,"score":0.2777000069618225},{"id":"https://openalex.org/C171752962","wikidata":"https://www.wikidata.org/wiki/Q255166","display_name":"Kullback\u2013Leibler divergence","level":2,"score":0.262800008058548},{"id":"https://openalex.org/C203858083","wikidata":"https://www.wikidata.org/wiki/Q188347","display_name":"Harmonic mean","level":2,"score":0.26269999146461487},{"id":"https://openalex.org/C182964748","wikidata":"https://www.wikidata.org/wiki/Q208216","display_name":"Triangle inequality","level":2,"score":0.26030001044273376},{"id":"https://openalex.org/C43555835","wikidata":"https://www.wikidata.org/wiki/Q2300258","display_name":"Conditional probability distribution","level":2,"score":0.25920000672340393},{"id":"https://openalex.org/C121194460","wikidata":"https://www.wikidata.org/wiki/Q856741","display_name":"Random walk","level":2,"score":0.2542000114917755},{"id":"https://openalex.org/C131770355","wikidata":"https://www.wikidata.org/wiki/Q876215","display_name":"Harmonic analysis","level":2,"score":0.25220000743865967}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.08517","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.08517","pdf_url":"https://arxiv.org/pdf/2610.08517","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.2610.08517","doi":"https://doi.org/10.48550/arxiv.2610.08517","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.08517","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:2610.08517","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.08517","pdf_url":"https://arxiv.org/pdf/2610.08517","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/W7220987380.pdf","grobid_xml":"https://content.openalex.org/works/W7220987380.grobid-xml"},"referenced_works_count":5,"referenced_works":["https://openalex.org/W3137426505","https://openalex.org/W7128742589","https://openalex.org/W7129260745","https://openalex.org/W7167218595","https://openalex.org/W7171471399"],"related_works":["https://openalex.org/W4392193692","https://openalex.org/W7171471399","https://openalex.org/W7167218595","https://openalex.org/W7129260745","https://openalex.org/W7221504818","https://openalex.org/W4411319843","https://openalex.org/W7140860576","https://openalex.org/W2520379570","https://openalex.org/W4399521719","https://openalex.org/W4385496405"],"abstract_inverted_index":{"We":[0,29,131],"study":[1],"the":[2,31,38,43,64,69,77,86,106,116,119,124,128,140,148,166,200],"random":[3],"geometric":[4],"graph":[5,40,125],"formed":[6],"by":[7,127,154],"connecting":[8],"$n$":[9],"independent":[10],"uniform":[11],"points":[12],"on":[13,76,123],"$\\mathbb":[14],"S^{d-1}$":[15],"whenever":[16],"their":[17],"inner":[18],"product":[19],"exceeds":[20],"a":[21,89,111,136,157,161,188,195],"threshold":[22,34],"chosen":[23],"to":[24,109,156],"give":[25],"edge":[26],"density":[27],"$p$.":[28],"establish":[30],"conjectured":[32],"dimensionality":[33],"for":[35,52,115,144],"indistinguishability":[36],"from":[37],"Erd\u00f6s-R\u00e9nyi":[39],"$G(n,p)$":[41],"throughout":[42],"sparse":[44],"regime":[45],"with":[46,94],"polynomially":[47],"growing":[48],"average":[49],"degree.":[50],"Specifically,":[51],"every":[53],"fixed":[54],"$\u03b4>0$,":[55],"if":[56],"$p\\to0$,":[57],"$p\\ge":[58],"n^{-1+\u03b4}$,":[59],"and":[60,96,185],"$d\\gg":[61],"(np\\log(1/p))^3$,":[62],"then":[63,132],"Kullback-Leibler":[65],"(KL)":[66],"divergence":[67],"between":[68],"two":[70],"distributions":[71],"vanishes.":[72],"Our":[73],"proof":[74],"builds":[75],"posterior":[78,172],"analysis":[79,93],"framework":[80],"introduced":[81],"in":[82,101,169],"previous":[83],"work":[84],"of":[85,118,139,171,174,191],"authors,":[87],"combining":[88],"truncated":[90,112],"second":[91],"moment":[92],"Fourier":[95,137],"spherical":[97,162],"harmonic":[98,163,176],"expansions.":[99],"As":[100],"prior":[102],"works,":[103],"we":[104],"reduce":[105],"KL":[107,202],"bound":[108,133],"controlling":[110],"$\u03c7^2$-divergence":[113],"$\u0394$":[114,134,186],"neighborhood":[117],"last":[120],"vertex":[121],"conditional":[122,141],"induced":[126],"preceding":[129],"vertices.":[130],"using":[135],"expansion":[138,164],"likelihood":[142],"ratio":[143],"this":[145],"neighborhood,":[146],"where":[147],"high-frequency":[149],"contribution":[150,168],"is":[151,181],"made":[152],"negligible":[153],"truncation":[155],"typical-degree":[158],"event,":[159],"while":[160],"approximates":[165],"low-frequency":[167],"terms":[170],"moments":[173,184],"bounded-degree":[175],"polynomials.":[177],"The":[178],"key":[179],"observation":[180],"that":[182,198],"these":[183],"satisfy":[187],"coupled":[189],"system":[190],"cavity":[192],"recurrences,":[193],"yielding":[194],"self-bounding":[196],"inequality":[197],"implies":[199],"desired":[201],"bound.":[203]},"counts_by_year":[{"year":2026,"cited_by_count":2}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-09T00:00:00"}