{"paper":{"title":"Galton-Watson trees with vanishing martingale limit","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Nadia Sidorova, Nathanael Berestycki, Nina Gantert, Peter Morters","submitted_at":"2012-04-13T19:20:12Z","abstract_excerpt":"We show that an infinite Galton-Watson tree, conditioned on its martingale limit being smaller than $\\eps$, agrees up to generation $K$ with a regular $\\mu$-ary tree, where $\\mu$ is the essential minimum of the offspring distribution and the random variable $K$ is strongly concentrated near an explicit deterministic function growing like a multiple of $\\log(1/\\eps)$. More precisely, we show that if $\\mu\\ge 2$ then with high probability as $\\eps \\downarrow 0$, $K$ takes exactly one or two values. This shows in particular that the conditioned trees converge to the regular $\\mu$-ary tree, providi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.3080","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"}