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Polynomials $\\{B_k\\}_{k=0}^{\\infty}$ are deterministic, and are selected from a standard basis such as Szeg\\H{o}, Bergman, or Faber polynomials associated with a Jordan domain $G$ bounded by an analytic curve. We show that the zero counting measures of $P_n$ converge almost surely to the equilibrium measure on the boundary of $G$ if and only if $\\mathbb{E}[\\log^+|A_0|]<\\infty$."},"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":"1607.02855","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2016-07-11T08:21:25Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"0a522bd78659e6f6285affffa1accabbdbb5746c69ba5289d55a815555b70815","abstract_canon_sha256":"24010fe6e8684f6ee8c3d0d3990d778a4142f601958f607796f0988b9b58b5af"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:11:14.540690Z","signature_b64":"Fk8rrXFUdaz2ZF789fAoQlWiQP5ycUxVLAavDJgMXk66hxv6M0kKkqgdF/0b9epiojsiuFhzKg29Zd9vruanDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"099c1634df3d1d2a7bb2a0ce753937c8a4652268f49c6e298ad83b913b08afa9","last_reissued_at":"2026-05-18T01:11:14.540231Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:11:14.540231Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Equidistribution of zeros of random polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CV","authors_text":"Igor Pritsker, Koushik Ramachandran","submitted_at":"2016-07-11T08:21:25Z","abstract_excerpt":"We study the asymptotic distribution of zeros for the random polynomials $P_n(z) = \\sum_{k=0}^n A_k B_k(z)$, where $\\{A_k\\}_{k=0}^{\\infty}$ are non-trivial i.i.d. complex random variables. Polynomials $\\{B_k\\}_{k=0}^{\\infty}$ are deterministic, and are selected from a standard basis such as Szeg\\H{o}, Bergman, or Faber polynomials associated with a Jordan domain $G$ bounded by an analytic curve. We show that the zero counting measures of $P_n$ converge almost surely to the equilibrium measure on the boundary of $G$ if and only if $\\mathbb{E}[\\log^+|A_0|]<\\infty$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.02855","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1607.02855","created_at":"2026-05-18T01:11:14.540303+00:00"},{"alias_kind":"arxiv_version","alias_value":"1607.02855v1","created_at":"2026-05-18T01:11:14.540303+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.02855","created_at":"2026-05-18T01:11:14.540303+00:00"},{"alias_kind":"pith_short_12","alias_value":"BGOBMNG7HUOS","created_at":"2026-05-18T12:30:07.202191+00:00"},{"alias_kind":"pith_short_16","alias_value":"BGOBMNG7HUOSU65S","created_at":"2026-05-18T12:30:07.202191+00:00"},{"alias_kind":"pith_short_8","alias_value":"BGOBMNG7","created_at":"2026-05-18T12:30:07.202191+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BGOBMNG7HUOSU65SUDHHKOJXZC","json":"https://pith.science/pith/BGOBMNG7HUOSU65SUDHHKOJXZC.json","graph_json":"https://pith.science/api/pith-number/BGOBMNG7HUOSU65SUDHHKOJXZC/graph.json","events_json":"https://pith.science/api/pith-number/BGOBMNG7HUOSU65SUDHHKOJXZC/events.json","paper":"https://pith.science/paper/BGOBMNG7"},"agent_actions":{"view_html":"https://pith.science/pith/BGOBMNG7HUOSU65SUDHHKOJXZC","download_json":"https://pith.science/pith/BGOBMNG7HUOSU65SUDHHKOJXZC.json","view_paper":"https://pith.science/paper/BGOBMNG7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1607.02855&json=true","fetch_graph":"https://pith.science/api/pith-number/BGOBMNG7HUOSU65SUDHHKOJXZC/graph.json","fetch_events":"https://pith.science/api/pith-number/BGOBMNG7HUOSU65SUDHHKOJXZC/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BGOBMNG7HUOSU65SUDHHKOJXZC/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BGOBMNG7HUOSU65SUDHHKOJXZC/action/storage_attestation","attest_author":"https://pith.science/pith/BGOBMNG7HUOSU65SUDHHKOJXZC/action/author_attestation","sign_citation":"https://pith.science/pith/BGOBMNG7HUOSU65SUDHHKOJXZC/action/citation_signature","submit_replication":"https://pith.science/pith/BGOBMNG7HUOSU65SUDHHKOJXZC/action/replication_record"}},"created_at":"2026-05-18T01:11:14.540303+00:00","updated_at":"2026-05-18T01:11:14.540303+00:00"}