{"paper":{"title":"Component structure of the configuration model: barely supercritical case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Malwina Luczak, Remco van der Hofstad, Svante Janson","submitted_at":"2016-11-17T15:11:00Z","abstract_excerpt":"We study near-critical behavior in the configuration model. Let $D_n$ be the degree of a random vertex. We let $\\nu_n={\\mathbb E} [D_n(D_n-1)]/{\\mathbb E}[D_n]$ and, assuming that $\\nu_n \\to 1$ as $n \\to \\infty$, we write $\\varepsilon_n=\\nu_n-1$. We call the setting where $\\varepsilon_n n^{1/3}/({\\mathbb E}[D_n^3])^{2/3} \\to \\infty$ the {\\it barely supercritical} regime. We further assume that the variance of $D_n$ is uniformly bounded as $n \\to \\infty$.\n  Let $D_n^*$ denote the size-biased version of $D_n$. We prove that there is a unique giant component of size $n \\rho_n {\\mathbb E} D_n (1+o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.05728","kind":"arxiv","version":1},"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"}