{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:QN7UYT7V6EEUQHPLLPPAL6F5JM","short_pith_number":"pith:QN7UYT7V","canonical_record":{"source":{"id":"1311.2907","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-11-12T20:20:23Z","cross_cats_sorted":[],"title_canon_sha256":"48acbe86ab72555cdbe4ef77ff6f1a20ef55a4348565a7922b2ef83248cb22e7","abstract_canon_sha256":"04030ab4fed60f5e8ad10f4ef9688d74443b82888b92e82298feee57bf3eafed"},"schema_version":"1.0"},"canonical_sha256":"837f4c4ff5f109481deb5bde05f8bd4b228d4e52270bf25a17fdbb3a81cd58c4","source":{"kind":"arxiv","id":"1311.2907","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.2907","created_at":"2026-05-18T01:23:39Z"},{"alias_kind":"arxiv_version","alias_value":"1311.2907v2","created_at":"2026-05-18T01:23:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.2907","created_at":"2026-05-18T01:23:39Z"},{"alias_kind":"pith_short_12","alias_value":"QN7UYT7V6EEU","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_16","alias_value":"QN7UYT7V6EEUQHPL","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_8","alias_value":"QN7UYT7V","created_at":"2026-05-18T12:27:57Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:QN7UYT7V6EEUQHPLLPPAL6F5JM","target":"record","payload":{"canonical_record":{"source":{"id":"1311.2907","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-11-12T20:20:23Z","cross_cats_sorted":[],"title_canon_sha256":"48acbe86ab72555cdbe4ef77ff6f1a20ef55a4348565a7922b2ef83248cb22e7","abstract_canon_sha256":"04030ab4fed60f5e8ad10f4ef9688d74443b82888b92e82298feee57bf3eafed"},"schema_version":"1.0"},"canonical_sha256":"837f4c4ff5f109481deb5bde05f8bd4b228d4e52270bf25a17fdbb3a81cd58c4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:23:39.033162Z","signature_b64":"A8pkD3PJ6kGlK7x8k1oeTxh5G1E2L8uu46g69OR1iXBd5oBYyvFQJeRPOFI+qOApaSy9GziNBGADW2LMP0UGAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"837f4c4ff5f109481deb5bde05f8bd4b228d4e52270bf25a17fdbb3a81cd58c4","last_reissued_at":"2026-05-18T01:23:39.032417Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:23:39.032417Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1311.2907","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":"2026-05-18T01:23:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ugU6RXGoF9dkeW8XIpAwF18evWv2PBmJU+cNfIh8pq3yc+miiY5wvVjboA+FPFjw5cm2Eplo2cHIE5Xmb+4lCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T14:16:29.116982Z"},"content_sha256":"491c483510d6cd6746ca56c9fdc22815fbb77f7f696760e07bc900d3b2035676","schema_version":"1.0","event_id":"sha256:491c483510d6cd6746ca56c9fdc22815fbb77f7f696760e07bc900d3b2035676"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:QN7UYT7V6EEUQHPLLPPAL6F5JM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Brownian Paths Homogeneously Distributed in Space: Percolation Phase Transition and Uniqueness of the Unbounded Cluster","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Dirk Erhard, Juli\\'an Mart\\'inez, Julien Poisat (CEREMADE)","submitted_at":"2013-11-12T20:20:23Z","abstract_excerpt":"We consider a continuum percolation model on $\\R^d$, $d\\geq 1$.For $t,\\lambda\\in (0,\\infty)$ and $d\\in\\{1,2,3\\}$, the occupied set is given by the union of independent Brownian paths running up to time $t$ whoseinitial points form a Poisson point process with intensity $\\lambda\\textgreater{}0$.When $d\\geq 4$, the Brownian paths are replaced by Wiener sausageswith radius $r\\textgreater{}0$.We establish that, for $d=1$ and all choices of $t$, no percolation occurs,whereas for $d\\geq 2$, there is a non-trivial percolation transitionin $t$, provided $\\lambda$ and $r$ are chosen properly.The last s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.2907","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":"2026-05-18T01:23:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qQlFiCWQXMeL0gcGnk5vAKlXO3+g4VOfP9Oclp4paMFb24jF5l7WZ9+m9exoSF4V7abluVJECz5iIEp3jNSdAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T14:16:29.117541Z"},"content_sha256":"0d639f7fae3998ef3e9049f72afe0187c38f3f51931dfbfc8ebfab6f355df704","schema_version":"1.0","event_id":"sha256:0d639f7fae3998ef3e9049f72afe0187c38f3f51931dfbfc8ebfab6f355df704"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QN7UYT7V6EEUQHPLLPPAL6F5JM/bundle.json","state_url":"https://pith.science/pith/QN7UYT7V6EEUQHPLLPPAL6F5JM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QN7UYT7V6EEUQHPLLPPAL6F5JM/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":"2026-06-08T14:16:29Z","links":{"resolver":"https://pith.science/pith/QN7UYT7V6EEUQHPLLPPAL6F5JM","bundle":"https://pith.science/pith/QN7UYT7V6EEUQHPLLPPAL6F5JM/bundle.json","state":"https://pith.science/pith/QN7UYT7V6EEUQHPLLPPAL6F5JM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QN7UYT7V6EEUQHPLLPPAL6F5JM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:QN7UYT7V6EEUQHPLLPPAL6F5JM","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":"04030ab4fed60f5e8ad10f4ef9688d74443b82888b92e82298feee57bf3eafed","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-11-12T20:20:23Z","title_canon_sha256":"48acbe86ab72555cdbe4ef77ff6f1a20ef55a4348565a7922b2ef83248cb22e7"},"schema_version":"1.0","source":{"id":"1311.2907","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.2907","created_at":"2026-05-18T01:23:39Z"},{"alias_kind":"arxiv_version","alias_value":"1311.2907v2","created_at":"2026-05-18T01:23:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.2907","created_at":"2026-05-18T01:23:39Z"},{"alias_kind":"pith_short_12","alias_value":"QN7UYT7V6EEU","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_16","alias_value":"QN7UYT7V6EEUQHPL","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_8","alias_value":"QN7UYT7V","created_at":"2026-05-18T12:27:57Z"}],"graph_snapshots":[{"event_id":"sha256:0d639f7fae3998ef3e9049f72afe0187c38f3f51931dfbfc8ebfab6f355df704","target":"graph","created_at":"2026-05-18T01:23:39Z","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 continuum percolation model on $\\R^d$, $d\\geq 1$.For $t,\\lambda\\in (0,\\infty)$ and $d\\in\\{1,2,3\\}$, the occupied set is given by the union of independent Brownian paths running up to time $t$ whoseinitial points form a Poisson point process with intensity $\\lambda\\textgreater{}0$.When $d\\geq 4$, the Brownian paths are replaced by Wiener sausageswith radius $r\\textgreater{}0$.We establish that, for $d=1$ and all choices of $t$, no percolation occurs,whereas for $d\\geq 2$, there is a non-trivial percolation transitionin $t$, provided $\\lambda$ and $r$ are chosen properly.The last s","authors_text":"Dirk Erhard, Juli\\'an Mart\\'inez, Julien Poisat (CEREMADE)","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-11-12T20:20:23Z","title":"Brownian Paths Homogeneously Distributed in Space: Percolation Phase Transition and Uniqueness of the Unbounded Cluster"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.2907","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":"sha256:491c483510d6cd6746ca56c9fdc22815fbb77f7f696760e07bc900d3b2035676","target":"record","created_at":"2026-05-18T01:23:39Z","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":"04030ab4fed60f5e8ad10f4ef9688d74443b82888b92e82298feee57bf3eafed","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-11-12T20:20:23Z","title_canon_sha256":"48acbe86ab72555cdbe4ef77ff6f1a20ef55a4348565a7922b2ef83248cb22e7"},"schema_version":"1.0","source":{"id":"1311.2907","kind":"arxiv","version":2}},"canonical_sha256":"837f4c4ff5f109481deb5bde05f8bd4b228d4e52270bf25a17fdbb3a81cd58c4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"837f4c4ff5f109481deb5bde05f8bd4b228d4e52270bf25a17fdbb3a81cd58c4","first_computed_at":"2026-05-18T01:23:39.032417Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:23:39.032417Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"A8pkD3PJ6kGlK7x8k1oeTxh5G1E2L8uu46g69OR1iXBd5oBYyvFQJeRPOFI+qOApaSy9GziNBGADW2LMP0UGAA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:23:39.033162Z","signed_message":"canonical_sha256_bytes"},"source_id":"1311.2907","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:491c483510d6cd6746ca56c9fdc22815fbb77f7f696760e07bc900d3b2035676","sha256:0d639f7fae3998ef3e9049f72afe0187c38f3f51931dfbfc8ebfab6f355df704"],"state_sha256":"e15661432cfb883d7038473292b123765601aeb61b5a231018a693cea876c4fc"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dG86tEjKRqrTFmEf9QUp5GaQpLQ8Inf8ypVXPF/raUua/B/7abGzccbh4HB3Jzbc1hP0+oeD6Y+GALZW/XhEBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T14:16:29.120160Z","bundle_sha256":"cd34f7f8f122db9ceca72a44b781679341fb6e20aa7f8a70c2c522d083368fab"}}