{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:VGUXTTIWVQNPVATEX44OKZIY5U","short_pith_number":"pith:VGUXTTIW","canonical_record":{"source":{"id":"1611.04567","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-11-14T20:44:16Z","cross_cats_sorted":[],"title_canon_sha256":"91ee2593a79752707bb699cc8f787fb87a59e59bdc11ba5c8a8e93027976630b","abstract_canon_sha256":"a04000b7da8a94549824164dfff0852fcdc7bb64151d2dbc83cc2730c065473f"},"schema_version":"1.0"},"canonical_sha256":"a9a979cd16ac1afa8264bf38e56518ed0b4ee10bb8bef2e1e1d2bb0eb42ac925","source":{"kind":"arxiv","id":"1611.04567","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.04567","created_at":"2026-05-18T00:53:47Z"},{"alias_kind":"arxiv_version","alias_value":"1611.04567v2","created_at":"2026-05-18T00:53:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.04567","created_at":"2026-05-18T00:53:47Z"},{"alias_kind":"pith_short_12","alias_value":"VGUXTTIWVQNP","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_16","alias_value":"VGUXTTIWVQNPVATE","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_8","alias_value":"VGUXTTIW","created_at":"2026-05-18T12:30:48Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:VGUXTTIWVQNPVATEX44OKZIY5U","target":"record","payload":{"canonical_record":{"source":{"id":"1611.04567","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-11-14T20:44:16Z","cross_cats_sorted":[],"title_canon_sha256":"91ee2593a79752707bb699cc8f787fb87a59e59bdc11ba5c8a8e93027976630b","abstract_canon_sha256":"a04000b7da8a94549824164dfff0852fcdc7bb64151d2dbc83cc2730c065473f"},"schema_version":"1.0"},"canonical_sha256":"a9a979cd16ac1afa8264bf38e56518ed0b4ee10bb8bef2e1e1d2bb0eb42ac925","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:53:47.582958Z","signature_b64":"AkC8/B8sTl5Q7SxNNh/BySPK4NedEjK86358fOP9eOiAvwN5HcSL+TYpHyj9j/jOqktV3FQvQeW4LREOzNooAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a9a979cd16ac1afa8264bf38e56518ed0b4ee10bb8bef2e1e1d2bb0eb42ac925","last_reissued_at":"2026-05-18T00:53:47.582503Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:53:47.582503Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1611.04567","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:53:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qPWr0n//Nu2YaT1QEDynxmYrPgvR7VN5JFNQlFpiQ6b34euq1IUN+ViKjEP6x9DJWuY8XN3tr1H2YcVLIH0IDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T20:18:11.016585Z"},"content_sha256":"6d0a48fc4324443ce63d67663a2aa7fb5b89114ccce4e1f2ab914df4c4300c4d","schema_version":"1.0","event_id":"sha256:6d0a48fc4324443ce63d67663a2aa7fb5b89114ccce4e1f2ab914df4c4300c4d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:VGUXTTIWVQNPVATEX44OKZIY5U","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Capacity of the range of random walk on $\\mathbb{Z}^4$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Amine Asselah, Bruno Schapira, Perla Sousi","submitted_at":"2016-11-14T20:44:16Z","abstract_excerpt":"We study the scaling limit of the capacity of the range of a simple random walk on the integer lattice in dimension four. We establish a strong law of large numbers and a central limit theorem with a non-gaussian limit. The asymptotic behaviour is analogous to that found by Le Gall in '86 for the volume of the range in dimension two."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.04567","kind":"arxiv","version":2},"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":"2026-05-18T00:53:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZU9K1fFyZ652BvR0HcNOIjMb1i9059Zuk2HH4wRJPJ3Pcm9AnMlRk5GO95O5RDamt9JkYpq2jtpw89Nt2bodCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T20:18:11.016916Z"},"content_sha256":"4cdc483ff99e2adc210fa67ce4d7da4d17eb2b8678933f01cee7736b69d3a361","schema_version":"1.0","event_id":"sha256:4cdc483ff99e2adc210fa67ce4d7da4d17eb2b8678933f01cee7736b69d3a361"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VGUXTTIWVQNPVATEX44OKZIY5U/bundle.json","state_url":"https://pith.science/pith/VGUXTTIWVQNPVATEX44OKZIY5U/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VGUXTTIWVQNPVATEX44OKZIY5U/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":"2026-06-20T20:18:11Z","links":{"resolver":"https://pith.science/pith/VGUXTTIWVQNPVATEX44OKZIY5U","bundle":"https://pith.science/pith/VGUXTTIWVQNPVATEX44OKZIY5U/bundle.json","state":"https://pith.science/pith/VGUXTTIWVQNPVATEX44OKZIY5U/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VGUXTTIWVQNPVATEX44OKZIY5U/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:VGUXTTIWVQNPVATEX44OKZIY5U","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":"a04000b7da8a94549824164dfff0852fcdc7bb64151d2dbc83cc2730c065473f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-11-14T20:44:16Z","title_canon_sha256":"91ee2593a79752707bb699cc8f787fb87a59e59bdc11ba5c8a8e93027976630b"},"schema_version":"1.0","source":{"id":"1611.04567","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.04567","created_at":"2026-05-18T00:53:47Z"},{"alias_kind":"arxiv_version","alias_value":"1611.04567v2","created_at":"2026-05-18T00:53:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.04567","created_at":"2026-05-18T00:53:47Z"},{"alias_kind":"pith_short_12","alias_value":"VGUXTTIWVQNP","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_16","alias_value":"VGUXTTIWVQNPVATE","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_8","alias_value":"VGUXTTIW","created_at":"2026-05-18T12:30:48Z"}],"graph_snapshots":[{"event_id":"sha256:4cdc483ff99e2adc210fa67ce4d7da4d17eb2b8678933f01cee7736b69d3a361","target":"graph","created_at":"2026-05-18T00:53:47Z","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 scaling limit of the capacity of the range of a simple random walk on the integer lattice in dimension four. We establish a strong law of large numbers and a central limit theorem with a non-gaussian limit. The asymptotic behaviour is analogous to that found by Le Gall in '86 for the volume of the range in dimension two.","authors_text":"Amine Asselah, Bruno Schapira, Perla Sousi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-11-14T20:44:16Z","title":"Capacity of the range of random walk on $\\mathbb{Z}^4$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.04567","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":"sha256:6d0a48fc4324443ce63d67663a2aa7fb5b89114ccce4e1f2ab914df4c4300c4d","target":"record","created_at":"2026-05-18T00:53:47Z","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":"a04000b7da8a94549824164dfff0852fcdc7bb64151d2dbc83cc2730c065473f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-11-14T20:44:16Z","title_canon_sha256":"91ee2593a79752707bb699cc8f787fb87a59e59bdc11ba5c8a8e93027976630b"},"schema_version":"1.0","source":{"id":"1611.04567","kind":"arxiv","version":2}},"canonical_sha256":"a9a979cd16ac1afa8264bf38e56518ed0b4ee10bb8bef2e1e1d2bb0eb42ac925","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a9a979cd16ac1afa8264bf38e56518ed0b4ee10bb8bef2e1e1d2bb0eb42ac925","first_computed_at":"2026-05-18T00:53:47.582503Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:53:47.582503Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"AkC8/B8sTl5Q7SxNNh/BySPK4NedEjK86358fOP9eOiAvwN5HcSL+TYpHyj9j/jOqktV3FQvQeW4LREOzNooAA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:53:47.582958Z","signed_message":"canonical_sha256_bytes"},"source_id":"1611.04567","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6d0a48fc4324443ce63d67663a2aa7fb5b89114ccce4e1f2ab914df4c4300c4d","sha256:4cdc483ff99e2adc210fa67ce4d7da4d17eb2b8678933f01cee7736b69d3a361"],"state_sha256":"ed4b8a936a154897f30f46f4c5b1f2fadd4a155607c37d48dabd81aa458b9b6a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rOWOTRL1eC3C0cHQ6WKel3DjPzyu2XnuB4Hz8h04UD11352guuXT4WahWGskV3lLX+92GaHYjkN8a2lHpsy8Bg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T20:18:11.018730Z","bundle_sha256":"45e56af1497218eca79d5f6f0c51d166412e2fba34f463e66f240552c10748ad"}}