{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T02:06:34Z","timestamp":1775527594856,"version":"3.50.1"},"reference-count":18,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T00:00:00Z","timestamp":1759276800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["J. Appl. Probab."],"published-print":{"date-parts":[[2026,3]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    We consider the number of edge crossings in a random graph drawing generated by projecting a random geometric graph on some compact convex set\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900225100302_inline1.png\"\/>\n                        <jats:tex-math>$W\\subset \\mathbb{R}^d$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    ,\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900225100302_inline2.png\"\/>\n                        <jats:tex-math>$d\\geq 3$<\/jats:tex-math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , onto a plane. The positions of these crossings form the support of a point process. We show that if the expected number of crossings converges to a positive but finite value, this point process converges to a Poisson point process in the Kantorovich\u2013Rubinstein distance. We further show a multivariate central limit theorem between the number of crossings and a second variable called the stress that holds when the expected vertex degree in the random geometric graph converges to a positive finite value.\n                  <\/jats:p>","DOI":"10.1017\/jpr.2025.10030","type":"journal-article","created":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T07:36:28Z","timestamp":1759304188000},"page":"258-281","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":0,"title":["Limit theorems for the number of crossings and stress in projections of a random geometric graph"],"prefix":"10.1017","volume":"63","author":[{"given":"Hanna","family":"D\u00f6ring","sequence":"first","affiliation":[{"name":"Osnabr\u00fcck University"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lianne","family":"De Jonge","sequence":"additional","affiliation":[{"name":"Osnabr\u00fcck University"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2025,10,1]]},"reference":[{"key":"S0021900225100302_ref3","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2018.04.007"},{"key":"S0021900225100302_ref15","doi-asserted-by":"crossref","unstructured":"[15] Schaefer, M. (2013). The graph crossing number and its variants: A survey. Electron. J. Combinatorics Dynamic Surveys, DS21.","DOI":"10.37236\/2713"},{"key":"S0021900225100302_ref9","doi-asserted-by":"publisher","DOI":"10.1239\/aap\/1401369698"},{"key":"S0021900225100302_ref13","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780198506263.001.0001"},{"key":"S0021900225100302_ref10","doi-asserted-by":"publisher","DOI":"10.1016\/j.endm.2011.10.009"},{"key":"S0021900225100302_ref12","first-page":"194","article-title":"Thirteen problems on crossing numbers","volume":"9","author":"Pach","year":"2000","journal-title":"Geombinatorics"},{"key":"S0021900225100302_ref4","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-04414-5_18"},{"key":"S0021900225100302_ref1","doi-asserted-by":"publisher","DOI":"10.1016\/j.spl.2023.109916"},{"key":"S0021900225100302_ref18","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.10053"},{"key":"S0021900225100302_ref2","unstructured":"[2] Arenas-Velilla, S. , Arizmendi, O. and Paguyo, J. E. (2023). Central limit theorem for crossings in randomly embedded graphs. Preprint, arXiv:2308.11570."},{"key":"S0021900225100302_ref14","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0021827"},{"key":"S0021900225100302_ref11","doi-asserted-by":"publisher","DOI":"10.1137\/0113032"},{"key":"S0021900225100302_ref16","doi-asserted-by":"publisher","DOI":"10.1007\/s10959-014-0576-6"},{"key":"S0021900225100302_ref5","unstructured":"[5] Chimani, M. , D\u00f6ring, H. and Reitzner, M. (2020). Crossing numbers and stress of random graphs. Preprint, arXiv:1808.07558."},{"key":"S0021900225100302_ref6","doi-asserted-by":"publisher","DOI":"10.1214\/15-AOP1020"},{"key":"S0021900225100302_ref8","doi-asserted-by":"publisher","DOI":"10.1017\/9781316104477"},{"key":"S0021900225100302_ref7","doi-asserted-by":"publisher","DOI":"10.1016\/j.comgeo.2019.04.003"},{"key":"S0021900225100302_ref17","doi-asserted-by":"publisher","DOI":"10.1214\/19-EJP386"}],"container-title":["Journal of Applied Probability"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0021900225100302","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T01:25:32Z","timestamp":1775525132000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0021900225100302\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,1]]},"references-count":18,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["S0021900225100302"],"URL":"https:\/\/doi.org\/10.1017\/jpr.2025.10030","relation":{},"ISSN":["0021-9002","1475-6072"],"issn-type":[{"value":"0021-9002","type":"print"},{"value":"1475-6072","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,10,1]]},"assertion":[{"value":"\u00a9 The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust","name":"copyright","label":"Copyright","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https:\/\/creativecommons.org\/licenses\/by\/4.0\/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.","name":"license","label":"License","group":{"name":"copyright_and_licensing","label":"Copyright and Licensing"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}