{"paper":{"title":"Range and critical generations of a random walk on Galton-Watson trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Pierre Andreoletti (MAPMO), Xinxin Chen (ICJ)","submitted_at":"2015-10-05T12:16:09Z","abstract_excerpt":"In this paper we consider a random walk in random environment on a tree and focus on the boundary case for the underlying branching potential. We study the range $R\\_n$ of this walk up to time $n$ and obtain its correct asymptotic in probability which is of order $n/\\log n$. Thisresult is a consequence of the asymptotical behavior of the number of visited sites at generations of order $(\\log n)^2$,which turn out to be the most visited generations. Our proof which involves a quenched analysisgives a description of the typical environments responsible for the behavior of $R\\_n$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.01121","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"}