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We study the following four families of random analytic functions: $\\sum_{k=0}^n \\sqrt{\\binom nk} \\xi_k z^k$ (spherical polynomials), $\\sum_{k=0}^\\infty \\sqrt{\\frac{n^k}{k!}} \\xi_k z^k$ (flat random analytic function), $\\sum_{k=0}^\\infty \\sqrt{\\binom {n+k-1} k} \\xi_k z^k$ (hyperbolic random analytic functions), $\\sum_{k=0}^n \\sqrt{\\frac{n^k}{k!}} \\xi_k z^k$ (Weyl polynomials). We compute explicitly the limiting mean density of real zeroes of these random functions. More precisely, we provide a formula for $\\"},"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":"1802.02390","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-02-07T11:34:49Z","cross_cats_sorted":[],"title_canon_sha256":"0f9a35c133884e15f7008f3edf0198bcf070c96e3c9d5f3036b51465fcf17da2","abstract_canon_sha256":"d9be7f13dadeca77ee434bb69b74eb6bb6bcced4e36b88e5c715a75efaea1a71"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:11:32.236268Z","signature_b64":"JzLrEDFYbXxtzh5cAAiuaoUuXeZEqmkmkeGyO0Wy+xMU9qAhL/KUMpk8nbS/hMw9YkNWINfBpifGun4lB4BeAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"269f06cfd903e84078c2aeacaeda19c8db7b573043ecabedf18dbe32f30675b5","last_reissued_at":"2026-05-18T00:11:32.235595Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:11:32.235595Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Real zeros of random analytic functions associated with geometries of constant curvature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Hendrik Flasche, Zakhar Kabluchko","submitted_at":"2018-02-07T11:34:49Z","abstract_excerpt":"Let $\\xi_0, \\xi_1, \\dots$ be i.i.d. random variables with zero mean and unit variance. We study the following four families of random analytic functions: $\\sum_{k=0}^n \\sqrt{\\binom nk} \\xi_k z^k$ (spherical polynomials), $\\sum_{k=0}^\\infty \\sqrt{\\frac{n^k}{k!}} \\xi_k z^k$ (flat random analytic function), $\\sum_{k=0}^\\infty \\sqrt{\\binom {n+k-1} k} \\xi_k z^k$ (hyperbolic random analytic functions), $\\sum_{k=0}^n \\sqrt{\\frac{n^k}{k!}} \\xi_k z^k$ (Weyl polynomials). We compute explicitly the limiting mean density of real zeroes of these random functions. 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