{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:WBEQ26Z7GXEIXT2PGYTRUISTPN","short_pith_number":"pith:WBEQ26Z7","canonical_record":{"source":{"id":"1503.01712","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-03-05T18:00:28Z","cross_cats_sorted":[],"title_canon_sha256":"ae3860bcea45a3232325fca28f9af8fbce52e50bbaebff77d7b32f0e7a507be0","abstract_canon_sha256":"f188ceea5b7803b0c824051a607bb1e9480fc9dd5a835c868ac7cb2054938166"},"schema_version":"1.0"},"canonical_sha256":"b0490d7b3f35c88bcf4f36271a22537b53804240cb4053b8d80ca4de00b08021","source":{"kind":"arxiv","id":"1503.01712","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.01712","created_at":"2026-05-17T23:39:37Z"},{"alias_kind":"arxiv_version","alias_value":"1503.01712v2","created_at":"2026-05-17T23:39:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.01712","created_at":"2026-05-17T23:39:37Z"},{"alias_kind":"pith_short_12","alias_value":"WBEQ26Z7GXEI","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"WBEQ26Z7GXEIXT2P","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"WBEQ26Z7","created_at":"2026-05-18T12:29:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:WBEQ26Z7GXEIXT2PGYTRUISTPN","target":"record","payload":{"canonical_record":{"source":{"id":"1503.01712","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-03-05T18:00:28Z","cross_cats_sorted":[],"title_canon_sha256":"ae3860bcea45a3232325fca28f9af8fbce52e50bbaebff77d7b32f0e7a507be0","abstract_canon_sha256":"f188ceea5b7803b0c824051a607bb1e9480fc9dd5a835c868ac7cb2054938166"},"schema_version":"1.0"},"canonical_sha256":"b0490d7b3f35c88bcf4f36271a22537b53804240cb4053b8d80ca4de00b08021","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:39:37.443761Z","signature_b64":"bVBhNEk7ieKqnO5EaIlMPrulGbwS8UtCcge2KLc5GL6Gl1vlV40sdyXIrZemjMDXYh3OBHi7zmXwszAmPZRCCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b0490d7b3f35c88bcf4f36271a22537b53804240cb4053b8d80ca4de00b08021","last_reissued_at":"2026-05-17T23:39:37.443112Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:39:37.443112Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1503.01712","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":"2026-05-17T23:39:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Yd/bAWP22YJo+/ohPvlLyL8hsNXs4uL2BFdYdaOwfvTtPxmCtVxld6NwjRvmJTHZTdWi+vzsm1aXOBIyhwYgCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T05:17:47.822200Z"},"content_sha256":"e73d50909b03db9fce4b2dd6aa87edc46837afc72a4f7e9eb6e2003b9cfe2c15","schema_version":"1.0","event_id":"sha256:e73d50909b03db9fce4b2dd6aa87edc46837afc72a4f7e9eb6e2003b9cfe2c15"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:WBEQ26Z7GXEIXT2PGYTRUISTPN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Asymptotics of the critical time in Wiener sausage percolation with a small radius","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Dirk Erhard, Julien Poisat (CEREMADE)","submitted_at":"2015-03-05T18:00:28Z","abstract_excerpt":"We consider a continuum percolation model on $\\R^d$, where $d\\geq 4$.The occupied set is given by the union of independent Wiener sausages with radius $r$ running up to time $t$ and whoseinitial points are distributed according to a homogeneous Poisson point process.It was established in a previous work by Erhard, Mart\\'{i}nez and Poisat~\\cite{EMP13} that (1) if $r$ is small enough there is a non-trivial percolation transitionin $t$ occuring at a critical time $t\\_c(r)$ and (2) in the supercritical regime the unbounded cluster is unique. In this paper we investigate the asymptotic behaviour of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.01712","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":"2026-05-17T23:39:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9waAUJQjD2qsiajndmr598WhnPHqjd4wmaFj1GrVKGhfJE1zYnUlrLSOEErjTpMT+0/FKhSZzaObvtO4/M37Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T05:17:47.822929Z"},"content_sha256":"a13c1865f31234adc648c54d8c268bf80a4feabf7fbd6514121ae8a741359791","schema_version":"1.0","event_id":"sha256:a13c1865f31234adc648c54d8c268bf80a4feabf7fbd6514121ae8a741359791"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WBEQ26Z7GXEIXT2PGYTRUISTPN/bundle.json","state_url":"https://pith.science/pith/WBEQ26Z7GXEIXT2PGYTRUISTPN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WBEQ26Z7GXEIXT2PGYTRUISTPN/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":"2026-06-08T05:17:47Z","links":{"resolver":"https://pith.science/pith/WBEQ26Z7GXEIXT2PGYTRUISTPN","bundle":"https://pith.science/pith/WBEQ26Z7GXEIXT2PGYTRUISTPN/bundle.json","state":"https://pith.science/pith/WBEQ26Z7GXEIXT2PGYTRUISTPN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WBEQ26Z7GXEIXT2PGYTRUISTPN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:WBEQ26Z7GXEIXT2PGYTRUISTPN","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":"f188ceea5b7803b0c824051a607bb1e9480fc9dd5a835c868ac7cb2054938166","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-03-05T18:00:28Z","title_canon_sha256":"ae3860bcea45a3232325fca28f9af8fbce52e50bbaebff77d7b32f0e7a507be0"},"schema_version":"1.0","source":{"id":"1503.01712","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.01712","created_at":"2026-05-17T23:39:37Z"},{"alias_kind":"arxiv_version","alias_value":"1503.01712v2","created_at":"2026-05-17T23:39:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.01712","created_at":"2026-05-17T23:39:37Z"},{"alias_kind":"pith_short_12","alias_value":"WBEQ26Z7GXEI","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"WBEQ26Z7GXEIXT2P","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"WBEQ26Z7","created_at":"2026-05-18T12:29:47Z"}],"graph_snapshots":[{"event_id":"sha256:a13c1865f31234adc648c54d8c268bf80a4feabf7fbd6514121ae8a741359791","target":"graph","created_at":"2026-05-17T23:39:37Z","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 $\\R^d$, where $d\\geq 4$.The occupied set is given by the union of independent Wiener sausages with radius $r$ running up to time $t$ and whoseinitial points are distributed according to a homogeneous Poisson point process.It was established in a previous work by Erhard, Mart\\'{i}nez and Poisat~\\cite{EMP13} that (1) if $r$ is small enough there is a non-trivial percolation transitionin $t$ occuring at a critical time $t\\_c(r)$ and (2) in the supercritical regime the unbounded cluster is unique. In this paper we investigate the asymptotic behaviour of","authors_text":"Dirk Erhard, Julien Poisat (CEREMADE)","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-03-05T18:00:28Z","title":"Asymptotics of the critical time in Wiener sausage percolation with a small radius"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.01712","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:e73d50909b03db9fce4b2dd6aa87edc46837afc72a4f7e9eb6e2003b9cfe2c15","target":"record","created_at":"2026-05-17T23:39:37Z","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":"f188ceea5b7803b0c824051a607bb1e9480fc9dd5a835c868ac7cb2054938166","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-03-05T18:00:28Z","title_canon_sha256":"ae3860bcea45a3232325fca28f9af8fbce52e50bbaebff77d7b32f0e7a507be0"},"schema_version":"1.0","source":{"id":"1503.01712","kind":"arxiv","version":2}},"canonical_sha256":"b0490d7b3f35c88bcf4f36271a22537b53804240cb4053b8d80ca4de00b08021","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b0490d7b3f35c88bcf4f36271a22537b53804240cb4053b8d80ca4de00b08021","first_computed_at":"2026-05-17T23:39:37.443112Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:39:37.443112Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"bVBhNEk7ieKqnO5EaIlMPrulGbwS8UtCcge2KLc5GL6Gl1vlV40sdyXIrZemjMDXYh3OBHi7zmXwszAmPZRCCg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:39:37.443761Z","signed_message":"canonical_sha256_bytes"},"source_id":"1503.01712","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e73d50909b03db9fce4b2dd6aa87edc46837afc72a4f7e9eb6e2003b9cfe2c15","sha256:a13c1865f31234adc648c54d8c268bf80a4feabf7fbd6514121ae8a741359791"],"state_sha256":"d828e739d2a40293644d299fb70be039fb6ff697452dc8c9819a1821bf09a44c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"R04eHRWhqsItL4MlZaK+1Pyx19WYZwUDAJsi+Cyn9f05XQD1bixMN4ssCRhKKg/NU1IRPHx/4+3p4L5oyYuiDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T05:17:47.826397Z","bundle_sha256":"421c8bbb2ad5a676a2185892dbb06b858d9e6808aff1a3ac7f8dc3bd79e5337b"}}