{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:3VZDRYJ5FAHGJ2M2XJWXWUMECK","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":"d205d303f360c129b742fee701420bf7834ecddf15c7ad6f0cfc2f2dc4b2ba6d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-11-30T19:28:09Z","title_canon_sha256":"c2bca7cdb18635452c5c6aaa485081d7fc1b54d79253e6c4bd8ba909a5b814f2"},"schema_version":"1.0","source":{"id":"1611.10324","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.10324","created_at":"2026-05-18T00:51:41Z"},{"alias_kind":"arxiv_version","alias_value":"1611.10324v2","created_at":"2026-05-18T00:51:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.10324","created_at":"2026-05-18T00:51:41Z"},{"alias_kind":"pith_short_12","alias_value":"3VZDRYJ5FAHG","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_16","alias_value":"3VZDRYJ5FAHGJ2M2","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_8","alias_value":"3VZDRYJ5","created_at":"2026-05-18T12:29:58Z"}],"graph_snapshots":[{"event_id":"sha256:56ab50c635fcc386c48f98aa79e6eac6b9c999ce4fe4b53772af7f831ff5b8cb","target":"graph","created_at":"2026-05-18T00:51:41Z","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 extend the theory of discrete capacity to critical branching random walk. We introduce branching capacity for any finite subset of $\\Z^d, d\\geq5$. Analogous to the regular discrete capacity, branching capacity is closely related to the asymptotics of the probability of visiting a fixed finite set by a critical branching random walk starting from a distant point and the conditional distribution of the hitting point.","authors_text":"Qingsan Zhu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-11-30T19:28:09Z","title":"On the critical branching random walk I: Branching capacity and visiting probability"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.10324","kind":"arxiv","version":2},"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":"sha256:41a89d9b3eca512a7b723617e11d134ffdfd55bf76f2adcaadf3bcfb706de997","target":"record","created_at":"2026-05-18T00:51:41Z","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":"d205d303f360c129b742fee701420bf7834ecddf15c7ad6f0cfc2f2dc4b2ba6d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-11-30T19:28:09Z","title_canon_sha256":"c2bca7cdb18635452c5c6aaa485081d7fc1b54d79253e6c4bd8ba909a5b814f2"},"schema_version":"1.0","source":{"id":"1611.10324","kind":"arxiv","version":2}},"canonical_sha256":"dd7238e13d280e64e99aba6d7b518412ab4cb565f2943794a573a5789dd828f4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dd7238e13d280e64e99aba6d7b518412ab4cb565f2943794a573a5789dd828f4","first_computed_at":"2026-05-18T00:51:41.922871Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:51:41.922871Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0eqHnHWE9sOMDABbIjZWz/MnTdMzzseC95MWjmlmeQSzWzmm0XNZ2ENvIoRPD01mdRprB87PwtFoNHGJA7WeDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:51:41.923508Z","signed_message":"canonical_sha256_bytes"},"source_id":"1611.10324","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:41a89d9b3eca512a7b723617e11d134ffdfd55bf76f2adcaadf3bcfb706de997","sha256:56ab50c635fcc386c48f98aa79e6eac6b9c999ce4fe4b53772af7f831ff5b8cb"],"state_sha256":"4400cc9129022e8a75bda09c7f67015caf264ac888e7fdcc6423a2e0523f1b33"}