{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:RTUWVTYVTNDMF7JSJ7RH6KPAO6","short_pith_number":"pith:RTUWVTYV","schema_version":"1.0","canonical_sha256":"8ce96acf159b46c2fd324fe27f29e077af7479c2666f1d2e61105bce2a25febe","source":{"kind":"arxiv","id":"1106.0354","version":1},"attestation_state":"computed","paper":{"title":"Scaling of cluster heterogeneity in percolation transitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cond-mat.stat-mech","authors_text":"Hyunggyu Park, Hyun Keun Lee, Jae Dong Noh","submitted_at":"2011-06-02T01:51:08Z","abstract_excerpt":"We investigate a critical scaling law for the cluster heterogeneity $H$ in site and bond percolations in $d$-dimensional lattices with $d=2,...,6$. The cluster heterogeneity is defined as the number of distinct cluster sizes. As an occupation probability $p$ increases, the cluster size distribution evolves from a monodisperse distribution to a polydisperse one in the subcritical phase, and back to a monodisperse one in the supercritical phase. We show analytically that $H$ diverges algebraically approaching the percolation critical point $p_c$ as $H\\sim |p-p_c|^{-1/\\sigma}$ with the critical e"},"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":"1106.0354","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.stat-mech","submitted_at":"2011-06-02T01:51:08Z","cross_cats_sorted":[],"title_canon_sha256":"ef9757711bb10b61f680ff3cbd65e669445495f3d025c3b993f7d9caa8f5b540","abstract_canon_sha256":"33f2d69cb8bb4cee130ca2da6ef01ec0e6ffea153155835bda1c1055263b71f9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:17:02.338454Z","signature_b64":"ziKGnOM1Txgd86HytoD5DzSOGqMYABDoZWbUV+cw7rnAAjuOZUPLiOvg5OqBEGDQ5yg7kVDk/DWIHgPDT2HVCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8ce96acf159b46c2fd324fe27f29e077af7479c2666f1d2e61105bce2a25febe","last_reissued_at":"2026-05-18T04:17:02.337767Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:17:02.337767Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Scaling of cluster heterogeneity in percolation transitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cond-mat.stat-mech","authors_text":"Hyunggyu Park, Hyun Keun Lee, Jae Dong Noh","submitted_at":"2011-06-02T01:51:08Z","abstract_excerpt":"We investigate a critical scaling law for the cluster heterogeneity $H$ in site and bond percolations in $d$-dimensional lattices with $d=2,...,6$. The cluster heterogeneity is defined as the number of distinct cluster sizes. As an occupation probability $p$ increases, the cluster size distribution evolves from a monodisperse distribution to a polydisperse one in the subcritical phase, and back to a monodisperse one in the supercritical phase. We show analytically that $H$ diverges algebraically approaching the percolation critical point $p_c$ as $H\\sim |p-p_c|^{-1/\\sigma}$ with the critical e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.0354","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1106.0354","created_at":"2026-05-18T04:17:02.337877+00:00"},{"alias_kind":"arxiv_version","alias_value":"1106.0354v1","created_at":"2026-05-18T04:17:02.337877+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1106.0354","created_at":"2026-05-18T04:17:02.337877+00:00"},{"alias_kind":"pith_short_12","alias_value":"RTUWVTYVTNDM","created_at":"2026-05-18T12:26:41.206345+00:00"},{"alias_kind":"pith_short_16","alias_value":"RTUWVTYVTNDMF7JS","created_at":"2026-05-18T12:26:41.206345+00:00"},{"alias_kind":"pith_short_8","alias_value":"RTUWVTYV","created_at":"2026-05-18T12:26:41.206345+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/RTUWVTYVTNDMF7JSJ7RH6KPAO6","json":"https://pith.science/pith/RTUWVTYVTNDMF7JSJ7RH6KPAO6.json","graph_json":"https://pith.science/api/pith-number/RTUWVTYVTNDMF7JSJ7RH6KPAO6/graph.json","events_json":"https://pith.science/api/pith-number/RTUWVTYVTNDMF7JSJ7RH6KPAO6/events.json","paper":"https://pith.science/paper/RTUWVTYV"},"agent_actions":{"view_html":"https://pith.science/pith/RTUWVTYVTNDMF7JSJ7RH6KPAO6","download_json":"https://pith.science/pith/RTUWVTYVTNDMF7JSJ7RH6KPAO6.json","view_paper":"https://pith.science/paper/RTUWVTYV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1106.0354&json=true","fetch_graph":"https://pith.science/api/pith-number/RTUWVTYVTNDMF7JSJ7RH6KPAO6/graph.json","fetch_events":"https://pith.science/api/pith-number/RTUWVTYVTNDMF7JSJ7RH6KPAO6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RTUWVTYVTNDMF7JSJ7RH6KPAO6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RTUWVTYVTNDMF7JSJ7RH6KPAO6/action/storage_attestation","attest_author":"https://pith.science/pith/RTUWVTYVTNDMF7JSJ7RH6KPAO6/action/author_attestation","sign_citation":"https://pith.science/pith/RTUWVTYVTNDMF7JSJ7RH6KPAO6/action/citation_signature","submit_replication":"https://pith.science/pith/RTUWVTYVTNDMF7JSJ7RH6KPAO6/action/replication_record"}},"created_at":"2026-05-18T04:17:02.337877+00:00","updated_at":"2026-05-18T04:17:02.337877+00:00"}