{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:XWBDUK3PZPV77O2MV6F3PQBCM7","short_pith_number":"pith:XWBDUK3P","canonical_record":{"source":{"id":"1309.5757","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-09-23T10:41:06Z","cross_cats_sorted":[],"title_canon_sha256":"635756a88be273ff8ec0322c084d6c5d374ee190ec68dfc6d7f9e7e5379c3b4f","abstract_canon_sha256":"29907f06d0996bdae9ff5742ad434908e58d52704bd83d1f7385e28c22d16942"},"schema_version":"1.0"},"canonical_sha256":"bd823a2b6fcbebffbb4caf8bb7c02267de81a07d57472fc6ae4a0387741df195","source":{"kind":"arxiv","id":"1309.5757","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.5757","created_at":"2026-05-18T02:25:49Z"},{"alias_kind":"arxiv_version","alias_value":"1309.5757v2","created_at":"2026-05-18T02:25:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.5757","created_at":"2026-05-18T02:25:49Z"},{"alias_kind":"pith_short_12","alias_value":"XWBDUK3PZPV7","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_16","alias_value":"XWBDUK3PZPV77O2M","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_8","alias_value":"XWBDUK3P","created_at":"2026-05-18T12:28:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:XWBDUK3PZPV77O2MV6F3PQBCM7","target":"record","payload":{"canonical_record":{"source":{"id":"1309.5757","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-09-23T10:41:06Z","cross_cats_sorted":[],"title_canon_sha256":"635756a88be273ff8ec0322c084d6c5d374ee190ec68dfc6d7f9e7e5379c3b4f","abstract_canon_sha256":"29907f06d0996bdae9ff5742ad434908e58d52704bd83d1f7385e28c22d16942"},"schema_version":"1.0"},"canonical_sha256":"bd823a2b6fcbebffbb4caf8bb7c02267de81a07d57472fc6ae4a0387741df195","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:25:49.289117Z","signature_b64":"a+g0kHUgb4LaLfSEfGjJjA+LPCpqqghuc0ZWLbuaZWI4qY1gRNCdkCBYU8Z3o5b3OS0RYt1JFx7l580aW6kGAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bd823a2b6fcbebffbb4caf8bb7c02267de81a07d57472fc6ae4a0387741df195","last_reissued_at":"2026-05-18T02:25:49.288694Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:25:49.288694Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1309.5757","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":"2026-05-18T02:25:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"y/Vc5tbn+G1cCQ0yWaB1BeApYcLv5zY8l96ZxST/SFTqOwGfjhdbJZu+kwcfzdAzNn7qd5/67gYETpr9XStUAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-21T20:09:16.344131Z"},"content_sha256":"2ba920216bc7db64cdae70ba193b4008fc343befacc63e649992d85c6b26135d","schema_version":"1.0","event_id":"sha256:2ba920216bc7db64cdae70ba193b4008fc343befacc63e649992d85c6b26135d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:XWBDUK3PZPV77O2MV6F3PQBCM7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Multiple phase transitions in long-range first-passage percolation on square lattices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Partha S. Dey, Shirshendu Chatterjee","submitted_at":"2013-09-23T10:41:06Z","abstract_excerpt":"We consider a model of long-range first-passage percolation on the $d$ dimensional square lattice $Z^d$ in which any two distinct vertices $x, y \\in Z^d$ are connected by an edge having exponentially distributed passage time with mean $||x-y||^{\\alpha+o(1)}$, where $\\alpha>0$ is a fixed parameter and $||\\cdot||$ is the $\\ell_1$-norm on $Z^d$. We analyze the asymptotic growth rate of the set $B_t$, which consists of all $x \\in Z^d$ such that the first-passage time between the origin 0 and $x$ is at most $t$, as $t\\to\\infty$. We show that depending on the values of $\\alpha$ there are four growth"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.5757","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":"2026-05-18T02:25:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hI/0fAmSGorbeBc9VNlHPRqdri5BU3yljeFeYfPix7Dy2D0uLRynFkV87jC32z29y5POFXqTu9UBQUWB7RvKCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-21T20:09:16.344512Z"},"content_sha256":"cd91d10c95b4523e1253203565845bf48a952fecec52c0a9b3dfec55a987d623","schema_version":"1.0","event_id":"sha256:cd91d10c95b4523e1253203565845bf48a952fecec52c0a9b3dfec55a987d623"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XWBDUK3PZPV77O2MV6F3PQBCM7/bundle.json","state_url":"https://pith.science/pith/XWBDUK3PZPV77O2MV6F3PQBCM7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XWBDUK3PZPV77O2MV6F3PQBCM7/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":"2026-05-21T20:09:16Z","links":{"resolver":"https://pith.science/pith/XWBDUK3PZPV77O2MV6F3PQBCM7","bundle":"https://pith.science/pith/XWBDUK3PZPV77O2MV6F3PQBCM7/bundle.json","state":"https://pith.science/pith/XWBDUK3PZPV77O2MV6F3PQBCM7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XWBDUK3PZPV77O2MV6F3PQBCM7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:XWBDUK3PZPV77O2MV6F3PQBCM7","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":"29907f06d0996bdae9ff5742ad434908e58d52704bd83d1f7385e28c22d16942","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-09-23T10:41:06Z","title_canon_sha256":"635756a88be273ff8ec0322c084d6c5d374ee190ec68dfc6d7f9e7e5379c3b4f"},"schema_version":"1.0","source":{"id":"1309.5757","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.5757","created_at":"2026-05-18T02:25:49Z"},{"alias_kind":"arxiv_version","alias_value":"1309.5757v2","created_at":"2026-05-18T02:25:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.5757","created_at":"2026-05-18T02:25:49Z"},{"alias_kind":"pith_short_12","alias_value":"XWBDUK3PZPV7","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_16","alias_value":"XWBDUK3PZPV77O2M","created_at":"2026-05-18T12:28:06Z"},{"alias_kind":"pith_short_8","alias_value":"XWBDUK3P","created_at":"2026-05-18T12:28:06Z"}],"graph_snapshots":[{"event_id":"sha256:cd91d10c95b4523e1253203565845bf48a952fecec52c0a9b3dfec55a987d623","target":"graph","created_at":"2026-05-18T02:25:49Z","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 consider a model of long-range first-passage percolation on the $d$ dimensional square lattice $Z^d$ in which any two distinct vertices $x, y \\in Z^d$ are connected by an edge having exponentially distributed passage time with mean $||x-y||^{\\alpha+o(1)}$, where $\\alpha>0$ is a fixed parameter and $||\\cdot||$ is the $\\ell_1$-norm on $Z^d$. We analyze the asymptotic growth rate of the set $B_t$, which consists of all $x \\in Z^d$ such that the first-passage time between the origin 0 and $x$ is at most $t$, as $t\\to\\infty$. We show that depending on the values of $\\alpha$ there are four growth","authors_text":"Partha S. Dey, Shirshendu Chatterjee","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-09-23T10:41:06Z","title":"Multiple phase transitions in long-range first-passage percolation on square lattices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.5757","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":"sha256:2ba920216bc7db64cdae70ba193b4008fc343befacc63e649992d85c6b26135d","target":"record","created_at":"2026-05-18T02:25:49Z","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":"29907f06d0996bdae9ff5742ad434908e58d52704bd83d1f7385e28c22d16942","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-09-23T10:41:06Z","title_canon_sha256":"635756a88be273ff8ec0322c084d6c5d374ee190ec68dfc6d7f9e7e5379c3b4f"},"schema_version":"1.0","source":{"id":"1309.5757","kind":"arxiv","version":2}},"canonical_sha256":"bd823a2b6fcbebffbb4caf8bb7c02267de81a07d57472fc6ae4a0387741df195","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bd823a2b6fcbebffbb4caf8bb7c02267de81a07d57472fc6ae4a0387741df195","first_computed_at":"2026-05-18T02:25:49.288694Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:25:49.288694Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"a+g0kHUgb4LaLfSEfGjJjA+LPCpqqghuc0ZWLbuaZWI4qY1gRNCdkCBYU8Z3o5b3OS0RYt1JFx7l580aW6kGAw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:25:49.289117Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.5757","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2ba920216bc7db64cdae70ba193b4008fc343befacc63e649992d85c6b26135d","sha256:cd91d10c95b4523e1253203565845bf48a952fecec52c0a9b3dfec55a987d623"],"state_sha256":"e55b6d07fe9c1f04ba296c262f1422659c972fa68d87ac961b9146d9d1c87969"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OpRPKJSJDuaJaj1+cuVdJS9ywnAFVmfo4fFNr0T7P65UNzfPm2R7xZx8L/BPPQHzBwRHQqtN9ZXS5X3Q9aStCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-21T20:09:16.346993Z","bundle_sha256":"3089623175ae8c5e71b1d76715d28edee6fc168c7e118b722446e5f5709df79f"}}