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Approximating $\\Pi$ by its restriction $\\Pi^n$ to $[n]\\:= \\{1, \\ldots, n\\},$ the suitably rescaled block counting process $n^{-1}\\#\\Pi^n(tn^{a-1})$ has a deterministic limit, $c(t)$, as $n\\to\\infty.$ An explicit formula for $c(t)$ is provided in Theorem 1.\n  The block size spectrum $(\\mathfrak{c}_{1}\\Pi^n(t), \\ldots, \\mathfrak{c}_{n}\\Pi^n(t)),$ where $\\mathfrak{c}_{i}\\Pi^n(t)$ counts the number of blocks of size $i$ in $\\Pi^n(t),$ captures more refined"},"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":"1611.06280","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-11-19T00:05:37Z","cross_cats_sorted":[],"title_canon_sha256":"8440033d6522cab84bb2f72a9b6420a08ee7f33534e9161ac07e89772286477b","abstract_canon_sha256":"360e74d215c6c29a55c48df81117dac314c0c1c5519e32c3912bc43a92821735"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:44:38.788297Z","signature_b64":"3qYFmouup7q4Q/sXKaywkRNicG0xcvv4TasIcdExQ8Oiopiy+qXProdpVHzK/Hg1Po0ncVrFGuI652UbM16RCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2a0d0eadb3a5d12e0c48d15ec59e5f9f46c4edf2b67d4fabb63e922e15503451","last_reissued_at":"2026-05-18T00:44:38.787565Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:44:38.787565Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The hydrodynamic limit of beta coalescents that come down from infinity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Helmut H. Pitters, Luke Miller","submitted_at":"2016-11-19T00:05:37Z","abstract_excerpt":"We quantify the manner in which the beta coalescent $\\Pi=\\{ \\Pi(t), t\\geq 0\\},$ with parameters $a\\in (0, 1),$ $b>0,$ comes down from infinity. Approximating $\\Pi$ by its restriction $\\Pi^n$ to $[n]\\:= \\{1, \\ldots, n\\},$ the suitably rescaled block counting process $n^{-1}\\#\\Pi^n(tn^{a-1})$ has a deterministic limit, $c(t)$, as $n\\to\\infty.$ An explicit formula for $c(t)$ is provided in Theorem 1.\n  The block size spectrum $(\\mathfrak{c}_{1}\\Pi^n(t), \\ldots, \\mathfrak{c}_{n}\\Pi^n(t)),$ where $\\mathfrak{c}_{i}\\Pi^n(t)$ counts the number of blocks of size $i$ in $\\Pi^n(t),$ captures more refined"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.06280","kind":"arxiv","version":2},"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":"1611.06280","created_at":"2026-05-18T00:44:38.787674+00:00"},{"alias_kind":"arxiv_version","alias_value":"1611.06280v2","created_at":"2026-05-18T00:44:38.787674+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.06280","created_at":"2026-05-18T00:44:38.787674+00:00"},{"alias_kind":"pith_short_12","alias_value":"FIGQ5LNTUXIS","created_at":"2026-05-18T12:30:15.759754+00:00"},{"alias_kind":"pith_short_16","alias_value":"FIGQ5LNTUXIS4DCI","created_at":"2026-05-18T12:30:15.759754+00:00"},{"alias_kind":"pith_short_8","alias_value":"FIGQ5LNT","created_at":"2026-05-18T12:30:15.759754+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/FIGQ5LNTUXIS4DCI2FPMLHS7T5","json":"https://pith.science/pith/FIGQ5LNTUXIS4DCI2FPMLHS7T5.json","graph_json":"https://pith.science/api/pith-number/FIGQ5LNTUXIS4DCI2FPMLHS7T5/graph.json","events_json":"https://pith.science/api/pith-number/FIGQ5LNTUXIS4DCI2FPMLHS7T5/events.json","paper":"https://pith.science/paper/FIGQ5LNT"},"agent_actions":{"view_html":"https://pith.science/pith/FIGQ5LNTUXIS4DCI2FPMLHS7T5","download_json":"https://pith.science/pith/FIGQ5LNTUXIS4DCI2FPMLHS7T5.json","view_paper":"https://pith.science/paper/FIGQ5LNT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1611.06280&json=true","fetch_graph":"https://pith.science/api/pith-number/FIGQ5LNTUXIS4DCI2FPMLHS7T5/graph.json","fetch_events":"https://pith.science/api/pith-number/FIGQ5LNTUXIS4DCI2FPMLHS7T5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FIGQ5LNTUXIS4DCI2FPMLHS7T5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FIGQ5LNTUXIS4DCI2FPMLHS7T5/action/storage_attestation","attest_author":"https://pith.science/pith/FIGQ5LNTUXIS4DCI2FPMLHS7T5/action/author_attestation","sign_citation":"https://pith.science/pith/FIGQ5LNTUXIS4DCI2FPMLHS7T5/action/citation_signature","submit_replication":"https://pith.science/pith/FIGQ5LNTUXIS4DCI2FPMLHS7T5/action/replication_record"}},"created_at":"2026-05-18T00:44:38.787674+00:00","updated_at":"2026-05-18T00:44:38.787674+00:00"}