{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,3]],"date-time":"2026-02-03T10:22:31Z","timestamp":1770114151510,"version":"3.49.0"},"reference-count":18,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2025,6,3]],"date-time":"2025-06-03T00:00:00Z","timestamp":1748908800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":["cambridge.org"],"crossmark-restriction":true},"short-container-title":["J. Appl. Probab."],"published-print":{"date-parts":[[2025,12]]},"abstract":"Abstract<\/jats:title>\n \n We define a random graph obtained by connecting each point of\n \n \n \n $\\mathbb{Z}^d$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n independently and uniformly to a fixed number\n \n \n \n $1 \\leq k \\leq 2d$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n of its nearest neighbors via a directed edge. We call this graph the directed\n k<\/jats:italic>\n -neighbor graph. Two natural associated undirected graphs are the undirected and the bidirectional\n k<\/jats:italic>\n -neighbor graph, where we connect two vertices by an undirected edge whenever there is a directed edge in the directed\n k<\/jats:italic>\n -neighbor graph between the vertices in at least one, respectively precisely two, directions. For these graphs we study the question of percolation, i.e. the existence of an infinite self-avoiding path. Using different kinds of proof techniques for different classes of cases, we show that for\n \n \n \n $k=1$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n even the undirected\n k<\/jats:italic>\n -neighbor graph never percolates, while the directed\n k<\/jats:italic>\n -neighbor graph percolates whenever\n \n \n \n $k \\geq d+1$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n ,\n \n \n \n $k \\geq 3$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , and\n \n \n \n $d \\geq 5$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , or\n \n \n \n $k \\geq 4$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and\n \n \n \n $d=4$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . We also show that the undirected 2-neighbor graph percolates for\n \n \n \n $d=2$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , the undirected 3-neighbor graph percolates for\n \n \n \n $d=3$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , and we provide some positive and negative percolation results regarding the bidirectional graph as well. A heuristic argument for high dimensions indicates that this class of models is a natural discrete analogue of the\n k<\/jats:italic>\n -nearest-neighbor graphs studied in continuum percolation, and our results support this interpretation.\n <\/jats:p>","DOI":"10.1017\/jpr.2025.15","type":"journal-article","created":{"date-parts":[[2025,6,3]],"date-time":"2025-06-03T01:56:52Z","timestamp":1748915812000},"page":"1475-1492","update-policy":"https:\/\/doi.org\/10.1017\/policypage","source":"Crossref","is-referenced-by-count":1,"title":["Percolation in lattice\n k<\/i>\n -neighbor graphs"],"prefix":"10.1017","volume":"62","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4212-0065","authenticated-orcid":false,"given":"Benedikt","family":"Jahnel","sequence":"first","affiliation":[{"name":"Technische Universit\u00e4t Braunschweig"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9188-1883","authenticated-orcid":false,"given":"Jonas","family":"K\u00f6ppl","sequence":"additional","affiliation":[{"name":"Weierstrass Institute for Applied Analysis and Stochastics"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5624-2410","authenticated-orcid":false,"given":"Bas","family":"Lodewijks","sequence":"additional","affiliation":[{"name":"Universit\u00e4t Augsburg"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4881-2553","authenticated-orcid":false,"given":"Andr\u00e1s","family":"T\u00f3bi\u00e1s","sequence":"additional","affiliation":[{"name":"Budapest University of Technology and Economics, and Alfr\u00e9d R\u00e9nyi Institute of Mathematics"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2025,6,3]]},"reference":[{"key":"S0021900225000154_ref14","doi-asserted-by":"publisher","DOI":"10.1090\/btran\/220"},{"key":"S0021900225000154_ref12","doi-asserted-by":"publisher","DOI":"10.30757\/ALEA.v18-26"},{"key":"S0021900225000154_ref1","unstructured":"[1] Amir, G. , Heydenreich, M. and Hirsch, C. (2024). Planar reinforced k-out percolation. Preprint, arXiv:2407.12484."},{"key":"S0021900225000154_ref3","doi-asserted-by":"publisher","DOI":"10.1214\/25-EJP1341"},{"key":"S0021900225000154_ref17","doi-asserted-by":"publisher","DOI":"10.37236\/1499"},{"key":"S0021900225000154_ref10","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20473"},{"key":"S0021900225000154_ref18","unstructured":"[18] Swart, J. (2017). A course in interacting particle systems 2017. Preprint, arXiv:1703.10007."},{"key":"S0021900225000154_ref15","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.21084"},{"key":"S0021900225000154_ref7","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1098-2418(199610)9:3<295::AID-RSA3>3.0.CO;2-S"},{"key":"S0021900225000154_ref5","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100060436"},{"key":"S0021900225000154_ref8","doi-asserted-by":"publisher","DOI":"10.1007\/BF01048021"},{"key":"S0021900225000154_ref2","first-page":"83","article-title":"Percolation in the k-nearest neighbor graph","author":"Balister","year":"2013","journal-title":"Recent Results in Designs and Graphs: A Tribute to"},{"key":"S0021900225000154_ref11","doi-asserted-by":"publisher","DOI":"10.1017\/S0963548316000158"},{"key":"S0021900225000154_ref4","unstructured":"[4] Coupier, D. , Henry, B. , Jahnel, B. and K\u00f6ppl, J. (2024). The planar lattice two-neighbor graph percolates. Preprint, arXiv:2412.20781."},{"key":"S0021900225000154_ref9","doi-asserted-by":"publisher","DOI":"10.1214\/20-AAP1587"},{"key":"S0021900225000154_ref16","volume-title":"Probability on Trees and Networks","author":"Lyons","year":"2017"},{"key":"S0021900225000154_ref6","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-32891-9"},{"key":"S0021900225000154_ref13","doi-asserted-by":"publisher","DOI":"10.1214\/20-AOP1476"}],"container-title":["Journal of Applied Probability"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0021900225000154","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,3]],"date-time":"2026-02-03T00:25:42Z","timestamp":1770078342000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0021900225000154\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,3]]},"references-count":18,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2025,12]]}},"alternative-id":["S0021900225000154"],"URL":"https:\/\/doi.org\/10.1017\/jpr.2025.15","relation":{},"ISSN":["0021-9002","1475-6072"],"issn-type":[{"value":"0021-9002","type":"print"},{"value":"1475-6072","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,6,3]]},"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"}}]}}