{"paper":{"title":"A first order phase transition in the threshold-$\\theta\\ge 2$ contact process on random $r$-regular graphs and $r$-trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Rick Durrett, Shirshendu Chatterjee","submitted_at":"2010-11-06T16:07:26Z","abstract_excerpt":"We consider the discrete-time threshold-$\\theta \\ge 2$ contact process on a random r-regular graph on n vertices. In this process, a vertex with at least \\theta occupied neighbors at time t will be occupied at time t+1 with probability p, and vacant otherwise. We show that if $\\theta \\ge 2$ and $r \\ge \\theta+2$, $\\epsilon_1$ is small and p is at least $p_1(\\epsilon_1)$, then starting from all vertices occupied the fraction of occupied vertices stays above $1-2\\epsilon_1$ up to time $\\exp(\\gamma_1(r)n)$ with probability at least $1 - \\exp(-\\gamma_1(r)n)$. In the other direction, we show that fo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.1567","kind":"arxiv","version":4},"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"}