{"paper":{"title":"A modified bootstrap percolation on a random graph coupled with a lattice","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Mikl\\'os Ruszink\\'o, Robert Kozma, Svante Janson, Yury Sokolov","submitted_at":"2015-07-29T00:47:34Z","abstract_excerpt":"In this paper a random graph model $G_{\\mathbb{Z}^2_N,p_d}$ is introduced, which is a combination of fixed torus grid edges in $(\\mathbb{Z}/N \\mathbb{Z})^2$ and some additional random ones. The random edges are called long, and the probability of having a long edge between vertices $u,v\\in(\\mathbb{Z}/N \\mathbb{Z})^2$ with graph distance $d$ on the torus grid is $p_d=c/Nd$, where $c$ is some constant. We show that, {\\em whp}, the diameter $D(G_{\\mathbb{Z}^2_N,p_d})=\\Theta (\\log N)$. Moreover, we consider non-monotonous bootstrap percolation on $G_{\\mathbb{Z}^2_N,p_d}$. We prove the presence of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.07997","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}