{"paper":{"title":"The thresholds for diameter 2 in random Cayley graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CO","authors_text":"Demetres Christofides, Klas Markstr\\\"om","submitted_at":"2011-08-17T18:45:41Z","abstract_excerpt":"Given a group G, the model $\\mathcal{G}(G,p)$ denotes the probability space of all Cayley graphs of G where each element of the generating set is chosen independently at random with probability p.\n  In this article we show that for any $\\epsilon > 0$ and any family of groups G_k of order n_k for which $n_k \\to \\infty$, a graph $\\Gamma_k \\in \\mathcal{G}(G_k,p)$ with high probability has diameter at most 2 if $p \\geqslant \\sqrt{(2 + \\epsilon) \\frac{\\log{n_k}}{n_k}}$ and with high probability has diameter greater than 2 if $p \\leqslant \\sqrt{(1/4 + \\epsilon)\\frac{\\log{n_k}}{n_k}}$.\n  We also prov"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.3547","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"}