{"id":"https://openalex.org/W7214079378","doi":"https://doi.org/10.48550/arxiv.2609.26230","title":"First-Order Laws for Random Geometric Graphs on the Torus","display_name":"First-Order Laws for Random Geometric Graphs on the Torus","publication_year":2026,"publication_date":"2026-08-12","ids":{"openalex":"https://openalex.org/W7214079378","doi":"https://doi.org/10.48550/arxiv.2609.26230"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.26230","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.26230","pdf_url":"https://arxiv.org/pdf/2609.26230","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/2609.26230","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5040390613","display_name":"Simi Haber","orcid":"https://orcid.org/0000-0001-8421-832X","observed_orcids":["https://orcid.org/0000-0001-8421-832X"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Haber, Simi","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5064287510","display_name":"Mostafa Mirabi","orcid":"https://orcid.org/0000-0003-0404-9904","observed_orcids":["https://orcid.org/0000-0003-0404-9904"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Mirabi, Mostafa","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5052112503","display_name":"Saharon shelah","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Shelah, Saharon","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":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/T11830","display_name":"Point processes and geometric inequalities","score":0.9092000126838684,"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"}},"topics":[{"id":"https://openalex.org/T11830","display_name":"Point processes and geometric inequalities","score":0.9092000126838684,"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/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.05469999834895134,"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":0.021700000390410423,"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/random-geometric-graphs","display_name":"random geometric graphs","score":0.996999979019165},{"id":"https://openalex.org/keywords/torus","display_name":"torus","score":0.9449999928474426},{"id":"https://openalex.org/keywords/zero-one-law","display_name":"zero-one law","score":0.7139999866485596},{"id":"https://openalex.org/keywords/convergence-law","display_name":"convergence law","score":0.8840000033378601},{"id":"https://openalex.org/keywords/twin-pairs","display_name":"twin pairs","score":0.8679999709129333},{"id":"https://openalex.org/keywords/poisson-limit","display_name":"Poisson limit","score":0.734000027179718},{"id":"https://openalex.org/keywords/origin-symmetric-convex-bodies","display_name":"origin-symmetric convex bodies","score":0.7390000224113464},{"id":"https://openalex.org/keywords/l-infinity-distance","display_name":"L-infinity distance","score":0.8029999732971191}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8550999760627747},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6547999978065491},{"id":"https://openalex.org/C9767117","wikidata":"https://www.wikidata.org/wiki/Q12510","display_name":"Torus","level":2,"score":0.6182000041007996},{"id":"https://openalex.org/C30609935","wikidata":"https://www.wikidata.org/wiki/Q7291969","display_name":"Random geometric graph","level":5,"score":0.5401999950408936},{"id":"https://openalex.org/C47458327","wikidata":"https://www.wikidata.org/wiki/Q910404","display_name":"Random graph","level":3,"score":0.5192999839782715},{"id":"https://openalex.org/C13355873","wikidata":"https://www.wikidata.org/wiki/Q2920850","display_name":"Connection (principal bundle)","level":2,"score":0.4627000093460083},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.44699999690055847},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.4262000024318695},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.4090000092983246},{"id":"https://openalex.org/C95763700","wikidata":"https://www.wikidata.org/wiki/Q578985","display_name":"Convergence of random variables","level":3,"score":0.40630000829696655},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.4059999883174896},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.4032000005245209},{"id":"https://openalex.org/C53471067","wikidata":"https://www.wikidata.org/wiki/Q7574076","display_name":"Spatial network","level":2,"score":0.3952000141143799},{"id":"https://openalex.org/C110484373","wikidata":"https://www.wikidata.org/wiki/Q264398","display_name":"Adjacency list","level":2,"score":0.39250001311302185},{"id":"https://openalex.org/C100906024","wikidata":"https://www.wikidata.org/wiki/Q205692","display_name":"Poisson distribution","level":2,"score":0.3357999920845032},{"id":"https://openalex.org/C7605037","wikidata":"https://www.wikidata.org/wiki/Q327164","display_name":"Geometric probability","level":2,"score":0.3319999873638153},{"id":"https://openalex.org/C134912446","wikidata":"https://www.wikidata.org/wiki/Q5166516","display_name":"Convex body","level":4,"score":0.3278999924659729},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.30799999833106995},{"id":"https://openalex.org/C185767445","wikidata":"https://www.wikidata.org/wiki/Q207952","display_name":"Law of large numbers","level":3,"score":0.2962000072002411},{"id":"https://openalex.org/C88230418","wikidata":"https://www.wikidata.org/wiki/Q131476","display_name":"Graph theory","level":2,"score":0.27230000495910645},{"id":"https://openalex.org/C122203268","wikidata":"https://www.wikidata.org/wiki/Q5862903","display_name":"Probability theory","level":2,"score":0.2711000144481659},{"id":"https://openalex.org/C16757284","wikidata":"https://www.wikidata.org/wiki/Q1145117","display_name":"Poisson point process","level":3,"score":0.26980000734329224},{"id":"https://openalex.org/C140532419","wikidata":"https://www.wikidata.org/wiki/Q249748","display_name":"Spectral radius","level":3,"score":0.2538999915122986}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.26230","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.26230","pdf_url":"https://arxiv.org/pdf/2609.26230","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.2609.26230","doi":"https://doi.org/10.48550/arxiv.2609.26230","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.26230","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:2609.26230","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.26230","pdf_url":"https://arxiv.org/pdf/2609.26230","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/G2432433410","display_name":null,"funder_award_id":"2320/23","funder_id":"https://openalex.org/F4320322252","funder_display_name":"Israel Science Foundation"},{"id":"https://openalex.org/G3110727907","display_name":null,"funder_award_id":"1838/19","funder_id":"https://openalex.org/F4320322252","funder_display_name":"Israel Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320322252","display_name":"Israel Science Foundation","ror":"https://ror.org/04sazxf24"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214079378.pdf","grobid_xml":"https://content.openalex.org/works/W7214079378.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W3173366859"],"related_works":["https://openalex.org/W7097676096","https://openalex.org/W1899503930","https://openalex.org/W7011071459","https://openalex.org/W2092678415","https://openalex.org/W7097281664","https://openalex.org/W2137034719","https://openalex.org/W2200393761","https://openalex.org/W4415188869","https://openalex.org/W2973137521","https://openalex.org/W2117753105"],"abstract_inverted_index":{"Let":[0],"$G_D(n;r)$":[1],"be":[2],"the":[3,14,45,48,70,82],"random":[4],"geometric":[5],"graph":[6],"generated":[7],"by":[8,20],"$n$":[9],"independent":[10],"uniform":[11],"points":[12],"on":[13],"$D$-dimensional":[15],"torus,":[16],"with":[17,78],"adjacency":[18],"defined":[19],"torus":[21],"$L^\\infty$-distance":[22],"at":[23,33],"most":[24],"$r$.":[25],"We":[26],"study":[27],"first-order":[28],"zero-one":[29,83],"and":[30,36],"convergence":[31],"laws":[32],"fixed":[34,41,88],"radius":[35],"in":[37,55],"sparse":[38],"regimes.":[39],"At":[40],"radius,":[42],"we":[43],"determine":[44],"asymptotics":[46],"of":[47,51],"expected":[49],"number":[50],"adjacent":[52],"twin":[53,71],"pairs":[54],"every":[56],"dimension.":[57],"In":[58],"dimension":[59],"two,":[60],"more":[61],"generally,":[62],"for":[63,86],"each":[64],"origin-symmetric":[65],"convex":[66],"connection":[67],"body":[68],"$K\\subset(-1/2,1/2)^2$,":[69],"count":[72],"converges":[73],"to":[74],"a":[75],"Poisson":[76],"variable":[77],"mean":[79],"$\\operatorname{area}(K^\\circ)/16$;":[80],"hence":[81],"law":[84],"fails":[85],"all":[87],"$0":[89]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-24T00:00:00"}