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Let $\\nu_n$ be the random measure assigning the same weight $1/n$ to each complex zero of $G_n$. Assuming essentially that $-\\frac 1n \\log f_{[tn], n}\\to u(t)$ as $n\\to\\infty$, where $u(t)$ is some function, we show that the measure $\\nu_n$ converges weakly to some deterministic measure which is characterized in t"},"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":"1205.5355","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-05-24T07:47:34Z","cross_cats_sorted":["math.CV"],"title_canon_sha256":"759564bedb204a616d964957c63fe8e90e3cf90a70f4845b2ecdc734c3b5eac7","abstract_canon_sha256":"4ae7a7596002d31e84e1c52d92139be52d2d1d46f50a00603aaf15de0c28e4f3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:44:24.023343Z","signature_b64":"py3DlTokFSnEObf8pqr3VR4Pw0niGpE0FFVFlHnM9yNCAxalDJA6gqh4qeubyYvneLoBWz7acme6jNyGXQ2xAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d83e41e4fd2f7efbd873f1ea42c0fbda9b4456a68bf285fe941315978c792bce","last_reissued_at":"2026-05-18T03:44:24.022789Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:44:24.022789Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Universality for zeros of random analytic functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.PR","authors_text":"Dmitry Zaporozhets, Zakhar Kabluchko","submitted_at":"2012-05-24T07:47:34Z","abstract_excerpt":"Let $\\xi_0,\\xi_1,...$ be independent identically distributed (i.i.d.) random variables such that $\\E \\log (1+|\\xi_0|)<\\infty$. We consider random analytic functions of the form $$ G_n(z)=\\sum_{k=0}^{\\infty} \\xi_k f_{k,n} z^k, $$ where $f_{k,n}$ are deterministic complex coefficients. Let $\\nu_n$ be the random measure assigning the same weight $1/n$ to each complex zero of $G_n$. Assuming essentially that $-\\frac 1n \\log f_{[tn], n}\\to u(t)$ as $n\\to\\infty$, where $u(t)$ is some function, we show that the measure $\\nu_n$ converges weakly to some deterministic measure which is characterized in t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.5355","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":"1205.5355","created_at":"2026-05-18T03:44:24.022878+00:00"},{"alias_kind":"arxiv_version","alias_value":"1205.5355v1","created_at":"2026-05-18T03:44:24.022878+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.5355","created_at":"2026-05-18T03:44:24.022878+00:00"},{"alias_kind":"pith_short_12","alias_value":"3A7EDZH5F57P","created_at":"2026-05-18T12:26:50.516681+00:00"},{"alias_kind":"pith_short_16","alias_value":"3A7EDZH5F57PXWDT","created_at":"2026-05-18T12:26:50.516681+00:00"},{"alias_kind":"pith_short_8","alias_value":"3A7EDZH5","created_at":"2026-05-18T12:26:50.516681+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/3A7EDZH5F57PXWDT6HVEFQH33K","json":"https://pith.science/pith/3A7EDZH5F57PXWDT6HVEFQH33K.json","graph_json":"https://pith.science/api/pith-number/3A7EDZH5F57PXWDT6HVEFQH33K/graph.json","events_json":"https://pith.science/api/pith-number/3A7EDZH5F57PXWDT6HVEFQH33K/events.json","paper":"https://pith.science/paper/3A7EDZH5"},"agent_actions":{"view_html":"https://pith.science/pith/3A7EDZH5F57PXWDT6HVEFQH33K","download_json":"https://pith.science/pith/3A7EDZH5F57PXWDT6HVEFQH33K.json","view_paper":"https://pith.science/paper/3A7EDZH5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1205.5355&json=true","fetch_graph":"https://pith.science/api/pith-number/3A7EDZH5F57PXWDT6HVEFQH33K/graph.json","fetch_events":"https://pith.science/api/pith-number/3A7EDZH5F57PXWDT6HVEFQH33K/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3A7EDZH5F57PXWDT6HVEFQH33K/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3A7EDZH5F57PXWDT6HVEFQH33K/action/storage_attestation","attest_author":"https://pith.science/pith/3A7EDZH5F57PXWDT6HVEFQH33K/action/author_attestation","sign_citation":"https://pith.science/pith/3A7EDZH5F57PXWDT6HVEFQH33K/action/citation_signature","submit_replication":"https://pith.science/pith/3A7EDZH5F57PXWDT6HVEFQH33K/action/replication_record"}},"created_at":"2026-05-18T03:44:24.022878+00:00","updated_at":"2026-05-18T03:44:24.022878+00:00"}