{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:EZVZBZCLBNMOXPATS7Q2U5Z4C6","short_pith_number":"pith:EZVZBZCL","canonical_record":{"source":{"id":"1011.4121","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-11-18T01:39:06Z","cross_cats_sorted":[],"title_canon_sha256":"8e55ec5a8c9123bb8749d677c67c3083aaacac7e25b227584be68ceeb112c6c7","abstract_canon_sha256":"e85504dd28f82d0d5bb5d5d3412c84cddc973198a9ea121977b9fc606e1a284d"},"schema_version":"1.0"},"canonical_sha256":"266b90e44b0b58ebbc1397e1aa773c17a626353e3fe844e247e9507ba59f4be1","source":{"kind":"arxiv","id":"1011.4121","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.4121","created_at":"2026-05-18T02:47:16Z"},{"alias_kind":"arxiv_version","alias_value":"1011.4121v1","created_at":"2026-05-18T02:47:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.4121","created_at":"2026-05-18T02:47:16Z"},{"alias_kind":"pith_short_12","alias_value":"EZVZBZCLBNMO","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_16","alias_value":"EZVZBZCLBNMOXPAT","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_8","alias_value":"EZVZBZCL","created_at":"2026-05-18T12:26:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:EZVZBZCLBNMOXPATS7Q2U5Z4C6","target":"record","payload":{"canonical_record":{"source":{"id":"1011.4121","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-11-18T01:39:06Z","cross_cats_sorted":[],"title_canon_sha256":"8e55ec5a8c9123bb8749d677c67c3083aaacac7e25b227584be68ceeb112c6c7","abstract_canon_sha256":"e85504dd28f82d0d5bb5d5d3412c84cddc973198a9ea121977b9fc606e1a284d"},"schema_version":"1.0"},"canonical_sha256":"266b90e44b0b58ebbc1397e1aa773c17a626353e3fe844e247e9507ba59f4be1","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:47:16.407379Z","signature_b64":"fnrh5NWqc3hZUxJi+syqcsbtg+bUu8tiVXe95xg4Z/gNLeXmPLAo+Na8/TOSwOuXxbPlfEiX6KGrG3gyvoBeAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"266b90e44b0b58ebbc1397e1aa773c17a626353e3fe844e247e9507ba59f4be1","last_reissued_at":"2026-05-18T02:47:16.406805Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:47:16.406805Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1011.4121","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:47:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BOfVFTklUSZ2TZVWDZShuxggXQ8azh4UQ1LTB1DYSCerVqVo2jWJggNKf5UHd2F2jUrPl3G2gUJl+c54AjDgAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T21:05:57.817244Z"},"content_sha256":"f057619573b6045d6b4b8577f8cba9cc8e335f0903d700dc3c4dc0f5c0b656cc","schema_version":"1.0","event_id":"sha256:f057619573b6045d6b4b8577f8cba9cc8e335f0903d700dc3c4dc0f5c0b656cc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:EZVZBZCLBNMOXPATS7Q2U5Z4C6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Sub-Gaussian tail bounds for the width and height of conditioned Galton--Watson trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Louigi Addario-Berry, Luc Devroye, Svante Janson","submitted_at":"2010-11-18T01:39:06Z","abstract_excerpt":"We study the height and width of a Galton--Watson tree with offspring distribution B satisfying E(B)=1, 0 < Var(B) < infinity, conditioned on having exactly n nodes. Under this conditioning, we derive sub-Gaussian tail bounds for both the width (largest number of nodes in any level) and height (greatest level containing a node); the bounds are optimal up to constant factors in the exponent. Under the same conditioning, we also derive essentially optimal upper tail bounds for the number of nodes at level k, for 1 <= k <= n."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.4121","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:47:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AImVUhFpOez06XXdujadTEfdxFiry8366tkeFaA2+MfKASUeiA+CPG1hC774zgETW0XudTd+JfMezmIUbyzVAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T21:05:57.817927Z"},"content_sha256":"167387126f04cf09389d5993e139dc1331a5c90f43bc1e4765806dc243d53e97","schema_version":"1.0","event_id":"sha256:167387126f04cf09389d5993e139dc1331a5c90f43bc1e4765806dc243d53e97"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EZVZBZCLBNMOXPATS7Q2U5Z4C6/bundle.json","state_url":"https://pith.science/pith/EZVZBZCLBNMOXPATS7Q2U5Z4C6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EZVZBZCLBNMOXPATS7Q2U5Z4C6/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-25T21:05:57Z","links":{"resolver":"https://pith.science/pith/EZVZBZCLBNMOXPATS7Q2U5Z4C6","bundle":"https://pith.science/pith/EZVZBZCLBNMOXPATS7Q2U5Z4C6/bundle.json","state":"https://pith.science/pith/EZVZBZCLBNMOXPATS7Q2U5Z4C6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EZVZBZCLBNMOXPATS7Q2U5Z4C6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:EZVZBZCLBNMOXPATS7Q2U5Z4C6","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"e85504dd28f82d0d5bb5d5d3412c84cddc973198a9ea121977b9fc606e1a284d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-11-18T01:39:06Z","title_canon_sha256":"8e55ec5a8c9123bb8749d677c67c3083aaacac7e25b227584be68ceeb112c6c7"},"schema_version":"1.0","source":{"id":"1011.4121","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.4121","created_at":"2026-05-18T02:47:16Z"},{"alias_kind":"arxiv_version","alias_value":"1011.4121v1","created_at":"2026-05-18T02:47:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.4121","created_at":"2026-05-18T02:47:16Z"},{"alias_kind":"pith_short_12","alias_value":"EZVZBZCLBNMO","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_16","alias_value":"EZVZBZCLBNMOXPAT","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_8","alias_value":"EZVZBZCL","created_at":"2026-05-18T12:26:06Z"}],"graph_snapshots":[{"event_id":"sha256:167387126f04cf09389d5993e139dc1331a5c90f43bc1e4765806dc243d53e97","target":"graph","created_at":"2026-05-18T02:47:16Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We study the height and width of a Galton--Watson tree with offspring distribution B satisfying E(B)=1, 0 < Var(B) < infinity, conditioned on having exactly n nodes. Under this conditioning, we derive sub-Gaussian tail bounds for both the width (largest number of nodes in any level) and height (greatest level containing a node); the bounds are optimal up to constant factors in the exponent. Under the same conditioning, we also derive essentially optimal upper tail bounds for the number of nodes at level k, for 1 <= k <= n.","authors_text":"Louigi Addario-Berry, Luc Devroye, Svante Janson","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-11-18T01:39:06Z","title":"Sub-Gaussian tail bounds for the width and height of conditioned Galton--Watson trees"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.4121","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:f057619573b6045d6b4b8577f8cba9cc8e335f0903d700dc3c4dc0f5c0b656cc","target":"record","created_at":"2026-05-18T02:47:16Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"e85504dd28f82d0d5bb5d5d3412c84cddc973198a9ea121977b9fc606e1a284d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-11-18T01:39:06Z","title_canon_sha256":"8e55ec5a8c9123bb8749d677c67c3083aaacac7e25b227584be68ceeb112c6c7"},"schema_version":"1.0","source":{"id":"1011.4121","kind":"arxiv","version":1}},"canonical_sha256":"266b90e44b0b58ebbc1397e1aa773c17a626353e3fe844e247e9507ba59f4be1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"266b90e44b0b58ebbc1397e1aa773c17a626353e3fe844e247e9507ba59f4be1","first_computed_at":"2026-05-18T02:47:16.406805Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:47:16.406805Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fnrh5NWqc3hZUxJi+syqcsbtg+bUu8tiVXe95xg4Z/gNLeXmPLAo+Na8/TOSwOuXxbPlfEiX6KGrG3gyvoBeAg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:47:16.407379Z","signed_message":"canonical_sha256_bytes"},"source_id":"1011.4121","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f057619573b6045d6b4b8577f8cba9cc8e335f0903d700dc3c4dc0f5c0b656cc","sha256:167387126f04cf09389d5993e139dc1331a5c90f43bc1e4765806dc243d53e97"],"state_sha256":"38c54d38f2bec17c6f30138b2102bce2a184d91aa93f11e96bbb18577b8ddb63"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"64QR2GYehxbbexQsuDTV96fD4T24ol2zEpQJDntWgvh+H3yQNMnLSKQEXwTbPXSAe3Dyl/5AKjeIZcPVfG6sBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T21:05:57.821306Z","bundle_sha256":"72b3d89d8c540ef7f654d7d426f0293ab4034bb3eb24504eed14b985bc84679d"}}