{"paper":{"title":"Mixing Time of Random Walk on Poisson Geometry Small World","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Xian-Yuan Wu","submitted_at":"2017-03-24T01:07:32Z","abstract_excerpt":"This paper focuses on the problem of modeling for small world effect on complex networks. Let's consider the supercritical Poisson continuous percolation on $d$-dimensional torus $T^d_n$ with volume $n^d$. By adding \"long edges (short cuts)\" randomly to the largest percolation cluster, we obtain a random graph $\\mathscr G_n$. In the present paper, we first prove that the diameter of $\\mathscr G_n$ grows at most polynomially fast in $\\ln n$ and we call it the Poisson Geometry Small World. Secondly, we prove that the random walk on $\\mathscr G_n$ possesses the rapid mixing property, namely, the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.08257","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"}