{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:STMNVGQQSXCXD2XGEH34XIYLKD","short_pith_number":"pith:STMNVGQQ","canonical_record":{"source":{"id":"1209.2038","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-10T15:38:37Z","cross_cats_sorted":["cs.DM","math.PR"],"title_canon_sha256":"62bc481231cbdef1d52808d5bb5e9d129e6c63b3fd43478ff4317748b0e80257","abstract_canon_sha256":"48d62966050bca004433c80f3058ea535fa30f7e22f5b39541e2b4fa3751343b"},"schema_version":"1.0"},"canonical_sha256":"94d8da9a1095c571eae621f7cba30b50f441b5ee27fe4b522daa575dd80a4717","source":{"kind":"arxiv","id":"1209.2038","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.2038","created_at":"2026-05-18T03:45:53Z"},{"alias_kind":"arxiv_version","alias_value":"1209.2038v1","created_at":"2026-05-18T03:45:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.2038","created_at":"2026-05-18T03:45:53Z"},{"alias_kind":"pith_short_12","alias_value":"STMNVGQQSXCX","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_16","alias_value":"STMNVGQQSXCXD2XG","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_8","alias_value":"STMNVGQQ","created_at":"2026-05-18T12:27:23Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:STMNVGQQSXCXD2XGEH34XIYLKD","target":"record","payload":{"canonical_record":{"source":{"id":"1209.2038","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-10T15:38:37Z","cross_cats_sorted":["cs.DM","math.PR"],"title_canon_sha256":"62bc481231cbdef1d52808d5bb5e9d129e6c63b3fd43478ff4317748b0e80257","abstract_canon_sha256":"48d62966050bca004433c80f3058ea535fa30f7e22f5b39541e2b4fa3751343b"},"schema_version":"1.0"},"canonical_sha256":"94d8da9a1095c571eae621f7cba30b50f441b5ee27fe4b522daa575dd80a4717","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:45:53.872365Z","signature_b64":"CarG1JvTLzoIZhtuvl015oII2ZjfacrnMtkg9MSNFARtRvpBFptTVDN5BF4bvvIS+1+usgbQcb9KhEz9clKmBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"94d8da9a1095c571eae621f7cba30b50f441b5ee27fe4b522daa575dd80a4717","last_reissued_at":"2026-05-18T03:45:53.871884Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:45:53.871884Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1209.2038","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":"2026-05-18T03:45:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"j573TmpOeXiRxTMI4IRIwb5UhJ4pv7T6zLmmYWPDarfJDUZfJEQY1xm4WhiZ9jrN9FAL8jlO5cEADFH/1+w+Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T01:43:32.596484Z"},"content_sha256":"979f38554ae2eebeb47f9b93d187fd10176f13216bf43fc0ff6b62fe380b973e","schema_version":"1.0","event_id":"sha256:979f38554ae2eebeb47f9b93d187fd10176f13216bf43fc0ff6b62fe380b973e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:STMNVGQQSXCXD2XGEH34XIYLKD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A natural stochastic extension of the sandpile model on a graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.PR"],"primary_cat":"math.CO","authors_text":"Jean-Fran\\c{c}ois Marckert, Thomas Selig, Yao-ban Chan","submitted_at":"2012-09-10T15:38:37Z","abstract_excerpt":"We introduce a new model of a stochastic sandpile on a graph $G$ containing a sink. When unstable, a site sends one grain to each of its neighbours independently with probability $p \\in (0,1]$. For $p=1$, this coincides with the standard Abelian sandpile model. In general, for $p\\in(0,1)$, the set of recurrent configurations of this sandpile model is different from that of the Abelian sandpile model. We give a characterisation of this set in terms of orientations of the graph $G$. We also define the lacking polynomial $L_G$ as the generating function counting this set according to the number o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.2038","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":"2026-05-18T03:45:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GYEfFdflx6P+4dj9N7uyKrFFhL2V5oN0uyQtp1A/asoMpTz2c59z6pRDJT9P9KmFOAkGJsjEHNMlsT3lcDXmCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T01:43:32.596855Z"},"content_sha256":"e72025d95d708689ad3711c27c8322c60677afee3f459b7bf681e9dfe895ed42","schema_version":"1.0","event_id":"sha256:e72025d95d708689ad3711c27c8322c60677afee3f459b7bf681e9dfe895ed42"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/STMNVGQQSXCXD2XGEH34XIYLKD/bundle.json","state_url":"https://pith.science/pith/STMNVGQQSXCXD2XGEH34XIYLKD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/STMNVGQQSXCXD2XGEH34XIYLKD/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":"2026-06-02T01:43:32Z","links":{"resolver":"https://pith.science/pith/STMNVGQQSXCXD2XGEH34XIYLKD","bundle":"https://pith.science/pith/STMNVGQQSXCXD2XGEH34XIYLKD/bundle.json","state":"https://pith.science/pith/STMNVGQQSXCXD2XGEH34XIYLKD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/STMNVGQQSXCXD2XGEH34XIYLKD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:STMNVGQQSXCXD2XGEH34XIYLKD","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":"48d62966050bca004433c80f3058ea535fa30f7e22f5b39541e2b4fa3751343b","cross_cats_sorted":["cs.DM","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-10T15:38:37Z","title_canon_sha256":"62bc481231cbdef1d52808d5bb5e9d129e6c63b3fd43478ff4317748b0e80257"},"schema_version":"1.0","source":{"id":"1209.2038","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.2038","created_at":"2026-05-18T03:45:53Z"},{"alias_kind":"arxiv_version","alias_value":"1209.2038v1","created_at":"2026-05-18T03:45:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.2038","created_at":"2026-05-18T03:45:53Z"},{"alias_kind":"pith_short_12","alias_value":"STMNVGQQSXCX","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_16","alias_value":"STMNVGQQSXCXD2XG","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_8","alias_value":"STMNVGQQ","created_at":"2026-05-18T12:27:23Z"}],"graph_snapshots":[{"event_id":"sha256:e72025d95d708689ad3711c27c8322c60677afee3f459b7bf681e9dfe895ed42","target":"graph","created_at":"2026-05-18T03:45:53Z","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 introduce a new model of a stochastic sandpile on a graph $G$ containing a sink. When unstable, a site sends one grain to each of its neighbours independently with probability $p \\in (0,1]$. For $p=1$, this coincides with the standard Abelian sandpile model. In general, for $p\\in(0,1)$, the set of recurrent configurations of this sandpile model is different from that of the Abelian sandpile model. We give a characterisation of this set in terms of orientations of the graph $G$. We also define the lacking polynomial $L_G$ as the generating function counting this set according to the number o","authors_text":"Jean-Fran\\c{c}ois Marckert, Thomas Selig, Yao-ban Chan","cross_cats":["cs.DM","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-10T15:38:37Z","title":"A natural stochastic extension of the sandpile model on a graph"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.2038","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":"sha256:979f38554ae2eebeb47f9b93d187fd10176f13216bf43fc0ff6b62fe380b973e","target":"record","created_at":"2026-05-18T03:45:53Z","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":"48d62966050bca004433c80f3058ea535fa30f7e22f5b39541e2b4fa3751343b","cross_cats_sorted":["cs.DM","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-10T15:38:37Z","title_canon_sha256":"62bc481231cbdef1d52808d5bb5e9d129e6c63b3fd43478ff4317748b0e80257"},"schema_version":"1.0","source":{"id":"1209.2038","kind":"arxiv","version":1}},"canonical_sha256":"94d8da9a1095c571eae621f7cba30b50f441b5ee27fe4b522daa575dd80a4717","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"94d8da9a1095c571eae621f7cba30b50f441b5ee27fe4b522daa575dd80a4717","first_computed_at":"2026-05-18T03:45:53.871884Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:45:53.871884Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CarG1JvTLzoIZhtuvl015oII2ZjfacrnMtkg9MSNFARtRvpBFptTVDN5BF4bvvIS+1+usgbQcb9KhEz9clKmBA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:45:53.872365Z","signed_message":"canonical_sha256_bytes"},"source_id":"1209.2038","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:979f38554ae2eebeb47f9b93d187fd10176f13216bf43fc0ff6b62fe380b973e","sha256:e72025d95d708689ad3711c27c8322c60677afee3f459b7bf681e9dfe895ed42"],"state_sha256":"6d6135e9955862464bbe550346411e74be21964bc435d5a30ac4d4cb34347b37"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6LB5j3OlRdJY1v8x32zpSk8Ge3lTD9YXs2/fPeoknTIPScAGW4NK2crIQlnJGyBUUpijXuBdtjvW32rdPtWjBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T01:43:32.598823Z","bundle_sha256":"c28e318c961a8706784516beb9a064931cabff9bb32ff09e9799d092f5c5c290"}}