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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1710.00937","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-10-02T23:10:15Z","cross_cats_sorted":["math.CV"],"title_canon_sha256":"f32689df071d4e00ecf858dff9ea8a7f9443344095ff26ad672edb6a02429b21","abstract_canon_sha256":"0cf9ad1ab1d0d08bddd4834e9021402a8d0841849947c9725353680658492c75"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:33:47.448104Z","signature_b64":"2G/fNPvUErxvImw8AyCp6AWjEoHL/ocaq0fcNtwsvyrHb3QNl9c7r7Gs4GpdExChgtjLYtzuHSdWgZcmUlrJDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f07aaa2cce680b981036c653253d19ff3d1106432718c4139265539163d70a80","last_reissued_at":"2026-05-18T00:33:47.447633Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:33:47.447633Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $G$. 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