{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:EPIYRDOZF2DTQTXPWLISVWM7VU","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":"cb372150369a14febd3f60a5f7a2c283caa8e7d8e057a86077a49609b7dd0b5c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-04-26T14:40:29Z","title_canon_sha256":"d7c849a93993943f361006ba595ffb44054e4cd0cd28dea2c35b9d924785066d"},"schema_version":"1.0","source":{"id":"1304.7190","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1304.7190","created_at":"2026-05-18T00:41:08Z"},{"alias_kind":"arxiv_version","alias_value":"1304.7190v4","created_at":"2026-05-18T00:41:08Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.7190","created_at":"2026-05-18T00:41:08Z"},{"alias_kind":"pith_short_12","alias_value":"EPIYRDOZF2DT","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"EPIYRDOZF2DTQTXP","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"EPIYRDOZ","created_at":"2026-05-18T12:27:43Z"}],"graph_snapshots":[{"event_id":"sha256:042462e765747ee4b67f5890099d8a0d49590bc323b5f221e76ae105b776d088","target":"graph","created_at":"2026-05-18T00:41:08Z","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 properties of the harmonic measure of balls in typical large discrete trees. For a ball of radius $n$ centered at the root, we prove that, although the size of the boundary is of order $n$, most of the harmonic measure is supported on a boundary set of size approximately equal to $n^{\\beta}$, where $\\beta\\approx0.78$ is a universal constant. To derive such results, we interpret harmonic measure as the exit distribution of the ball by simple random walk on the tree, and we first deal with the case of critical Galton-Watson trees conditioned to have height greater than $n$. An important","authors_text":"Jean-Fran\\c{c}ois Le Gall, Nicolas Curien","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-04-26T14:40:29Z","title":"The harmonic measure of balls in random trees"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.7190","kind":"arxiv","version":4},"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":"sha256:31e21d89f25ad5bb521bf5f6333a35f6175ce00ddeb0b0c52d010e062aa88d53","target":"record","created_at":"2026-05-18T00:41:08Z","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":"cb372150369a14febd3f60a5f7a2c283caa8e7d8e057a86077a49609b7dd0b5c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-04-26T14:40:29Z","title_canon_sha256":"d7c849a93993943f361006ba595ffb44054e4cd0cd28dea2c35b9d924785066d"},"schema_version":"1.0","source":{"id":"1304.7190","kind":"arxiv","version":4}},"canonical_sha256":"23d1888dd92e87384eefb2d12ad99fad06d3b68b89e33b722afea259f1c4cfc5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"23d1888dd92e87384eefb2d12ad99fad06d3b68b89e33b722afea259f1c4cfc5","first_computed_at":"2026-05-18T00:41:08.672270Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:41:08.672270Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"g7ncy9MuOCX8Q3a1ACklmix2SVjUntSI+ySZaS+6HqPKx7zXYMxd6IkzAiTLL7DXn2W+VmI7jNm+vdWd4nxJCA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:41:08.672826Z","signed_message":"canonical_sha256_bytes"},"source_id":"1304.7190","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:31e21d89f25ad5bb521bf5f6333a35f6175ce00ddeb0b0c52d010e062aa88d53","sha256:042462e765747ee4b67f5890099d8a0d49590bc323b5f221e76ae105b776d088"],"state_sha256":"6f44d068a87f09845ceb7240f54f5b0ed04d16cd4efb6029d7a7b8f669905f01"}