{"paper":{"title":"Biased random walks on a Galton-Watson tree with leaves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Alan Hammond, Alexander Fribergh (CIMS), G\\'erard Ben Arous (CIMS), Nina Gantert","submitted_at":"2007-11-23T09:37:11Z","abstract_excerpt":"We consider a biased random walk $X_n$ on a Galton-Watson tree with leaves in the sub-ballistic regime. We prove that there exists an explicit constant $\\gamma= \\gamma(\\beta) \\in (0,1)$, depending on the bias $\\beta$, such that $X_n$ is of order $n^{\\gamma}$. Denoting $\\Delta_n$ the hitting time of level $n$, we prove that $\\Delta_n/n^{1/\\gamma}$ is tight. Moreover we show that $\\Delta_n/n^{1/\\gamma}$ does not converge in law (at least for large values of $\\beta$). We prove that along the sequences $n_{\\lambda}(k)=\\lfloor \\lambda \\beta^{\\gamma k}\\rfloor$, $\\Delta_n/n^{1/\\gamma}$ converges to c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0711.3686","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"}