{"paper":{"title":"Natural boundary and zero distribution of random polynomials in smooth domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.PR","authors_text":"Igor Pritsker, Koushik Ramachandran","submitted_at":"2017-10-02T23:10:15Z","abstract_excerpt":"We consider the zero distribution of random polynomials of the form $P_n(z) = \\sum_{k=0}^n a_k B_k(z)$, where $\\{a_k\\}_{k=0}^{\\infty}$ are non-trivial i.i.d. complex random variables with mean $0$ and finite variance. Polynomials $\\{B_k\\}_{k=0}^{\\infty}$ are selected from a standard basis such as Szeg\\H{o}, Bergman, or Faber polynomials associated with a Jordan domain $G$ whose boundary is $C^{2, \\alpha}$ smooth. We show that the zero counting measures of $P_n$ converge almost surely to the equilibrium measure on the boundary of $G$. We also show that if $\\{a_k\\}_{k=0}^{\\infty}$ are i.i.d. ran"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.00937","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"}