{"paper":{"title":"Quenched Survival of Bernoulli Percolation on Galton-Watson Trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Josh Rosenberg, Marcus Michelen, Robin Pemantle","submitted_at":"2018-05-09T18:52:59Z","abstract_excerpt":"We explore the survival function for percolation on Galton-Watson trees. Letting $g(T,p)$ represent the probability a tree $T$ survives Bernoulli percolation with parameter $p$, we establish several results about the behavior of the random function $g(\\mathbf{T} , \\cdot)$, where $\\mathbf{T}$ is drawn from the Galton-Watson distribution. These include almost sure smoothness in the supercritical region; an expression for the $k\\text{th}$-order Taylor expansion of $g(\\mathbf{T} , \\cdot)$ at criticality in terms of limits of martingales defined from $\\mathbf{T}$ (this requires a moment condition d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.03693","kind":"arxiv","version":2},"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"}