{"paper":{"title":"Block size in Geometric(p)-biased permutations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Erin Beckman, Irina Cristali, Jake Steinberg, James Nolen, Matthew Junge, Rick Durrett, Vinit Ranjan","submitted_at":"2017-08-18T14:23:57Z","abstract_excerpt":"Fix a probability distribution $\\mathbf p = (p_1, p_2, \\cdots)$ on the positive integers. The first block in a $\\mathbf p$-biased permutation can be visualized in terms of raindrops that land at each positive integer $j$ with probability $p_j$. It is the first point $K$ so that all sites in $[1,K]$ are wet and all sites in $(K,\\infty)$ are dry. For the geometric distribution $p_j= p(1-p)^{j-1}$ we show that $p \\log K$ converges in probability to an explicit constant as $p$ tends to 0. Additionally, we prove that if $\\mathbf p$ has a stretch exponential distribution, then $K$ is infinite with p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.05626","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"}