{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:34EKX3ZA64ZR3NMUCGX3YNLVES","short_pith_number":"pith:34EKX3ZA","canonical_record":{"source":{"id":"1011.1567","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-11-06T16:07:26Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"403227be97bd85a69085b180348aff7c16ce81b624c88e989b919f2ae812c912","abstract_canon_sha256":"755ab1bcbd135b9f1d91abc18a3f23982ab58bab9e2c68986a62304a32cd7a60"},"schema_version":"1.0"},"canonical_sha256":"df08abef20f7331db59411afbc357524a4be8e8241e272662489b51409cf3b7b","source":{"kind":"arxiv","id":"1011.1567","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.1567","created_at":"2026-05-18T03:10:14Z"},{"alias_kind":"arxiv_version","alias_value":"1011.1567v4","created_at":"2026-05-18T03:10:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.1567","created_at":"2026-05-18T03:10:14Z"},{"alias_kind":"pith_short_12","alias_value":"34EKX3ZA64ZR","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_16","alias_value":"34EKX3ZA64ZR3NMU","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_8","alias_value":"34EKX3ZA","created_at":"2026-05-18T12:26:03Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:34EKX3ZA64ZR3NMUCGX3YNLVES","target":"record","payload":{"canonical_record":{"source":{"id":"1011.1567","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-11-06T16:07:26Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"403227be97bd85a69085b180348aff7c16ce81b624c88e989b919f2ae812c912","abstract_canon_sha256":"755ab1bcbd135b9f1d91abc18a3f23982ab58bab9e2c68986a62304a32cd7a60"},"schema_version":"1.0"},"canonical_sha256":"df08abef20f7331db59411afbc357524a4be8e8241e272662489b51409cf3b7b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:10:14.797138Z","signature_b64":"U4qcVtAmDY5WeUiecBAQRG4R8E1NumFOjw7EsuT/dfma67j7TvSuLCaWjtQy+9SEzpVYUC2EjX3gkQezjuiPBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"df08abef20f7331db59411afbc357524a4be8e8241e272662489b51409cf3b7b","last_reissued_at":"2026-05-18T03:10:14.796431Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:10:14.796431Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1011.1567","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:10:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SlH69fnN8gqExDWu9nH/RZ2ZAM/cBD/sBslTBTLsJLIElcBpt+Yi3eU7jVEU0o/5rWrEM92Zq6aRgxoXEqroAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T11:58:11.210838Z"},"content_sha256":"ec85d8f7faa4926e4c5916b8d8b3226c32fef0ce20bd67f945d3078e7ae8c4c5","schema_version":"1.0","event_id":"sha256:ec85d8f7faa4926e4c5916b8d8b3226c32fef0ce20bd67f945d3078e7ae8c4c5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:34EKX3ZA64ZR3NMUCGX3YNLVES","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A first order phase transition in the threshold-$\\theta\\ge 2$ contact process on random $r$-regular graphs and $r$-trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Rick Durrett, Shirshendu Chatterjee","submitted_at":"2010-11-06T16:07:26Z","abstract_excerpt":"We consider the discrete-time threshold-$\\theta \\ge 2$ contact process on a random r-regular graph on n vertices. In this process, a vertex with at least \\theta occupied neighbors at time t will be occupied at time t+1 with probability p, and vacant otherwise. We show that if $\\theta \\ge 2$ and $r \\ge \\theta+2$, $\\epsilon_1$ is small and p is at least $p_1(\\epsilon_1)$, then starting from all vertices occupied the fraction of occupied vertices stays above $1-2\\epsilon_1$ up to time $\\exp(\\gamma_1(r)n)$ with probability at least $1 - \\exp(-\\gamma_1(r)n)$. In the other direction, we show that fo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.1567","kind":"arxiv","version":4},"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":"2026-05-18T03:10:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ebB+VmlE19BxjFVGN6aBQopW6PidCDQvAVTspMEr1gQMPGSB9BHMuZrJNQ0AS/AgQMTf1U26WD9KuX9wjTzPCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T11:58:11.211591Z"},"content_sha256":"c409325cbe6c0259805b2c89ebbf380a8af9a887d96d24423846483ee0505252","schema_version":"1.0","event_id":"sha256:c409325cbe6c0259805b2c89ebbf380a8af9a887d96d24423846483ee0505252"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/34EKX3ZA64ZR3NMUCGX3YNLVES/bundle.json","state_url":"https://pith.science/pith/34EKX3ZA64ZR3NMUCGX3YNLVES/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/34EKX3ZA64ZR3NMUCGX3YNLVES/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":"2026-05-27T11:58:11Z","links":{"resolver":"https://pith.science/pith/34EKX3ZA64ZR3NMUCGX3YNLVES","bundle":"https://pith.science/pith/34EKX3ZA64ZR3NMUCGX3YNLVES/bundle.json","state":"https://pith.science/pith/34EKX3ZA64ZR3NMUCGX3YNLVES/state.json","well_known_bundle":"https://pith.science/.well-known/pith/34EKX3ZA64ZR3NMUCGX3YNLVES/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:34EKX3ZA64ZR3NMUCGX3YNLVES","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":"755ab1bcbd135b9f1d91abc18a3f23982ab58bab9e2c68986a62304a32cd7a60","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-11-06T16:07:26Z","title_canon_sha256":"403227be97bd85a69085b180348aff7c16ce81b624c88e989b919f2ae812c912"},"schema_version":"1.0","source":{"id":"1011.1567","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1011.1567","created_at":"2026-05-18T03:10:14Z"},{"alias_kind":"arxiv_version","alias_value":"1011.1567v4","created_at":"2026-05-18T03:10:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.1567","created_at":"2026-05-18T03:10:14Z"},{"alias_kind":"pith_short_12","alias_value":"34EKX3ZA64ZR","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_16","alias_value":"34EKX3ZA64ZR3NMU","created_at":"2026-05-18T12:26:03Z"},{"alias_kind":"pith_short_8","alias_value":"34EKX3ZA","created_at":"2026-05-18T12:26:03Z"}],"graph_snapshots":[{"event_id":"sha256:c409325cbe6c0259805b2c89ebbf380a8af9a887d96d24423846483ee0505252","target":"graph","created_at":"2026-05-18T03:10:14Z","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 the discrete-time threshold-$\\theta \\ge 2$ contact process on a random r-regular graph on n vertices. In this process, a vertex with at least \\theta occupied neighbors at time t will be occupied at time t+1 with probability p, and vacant otherwise. We show that if $\\theta \\ge 2$ and $r \\ge \\theta+2$, $\\epsilon_1$ is small and p is at least $p_1(\\epsilon_1)$, then starting from all vertices occupied the fraction of occupied vertices stays above $1-2\\epsilon_1$ up to time $\\exp(\\gamma_1(r)n)$ with probability at least $1 - \\exp(-\\gamma_1(r)n)$. In the other direction, we show that fo","authors_text":"Rick Durrett, Shirshendu Chatterjee","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-11-06T16:07:26Z","title":"A first order phase transition in the threshold-$\\theta\\ge 2$ contact process on random $r$-regular graphs and $r$-trees"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.1567","kind":"arxiv","version":4},"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:ec85d8f7faa4926e4c5916b8d8b3226c32fef0ce20bd67f945d3078e7ae8c4c5","target":"record","created_at":"2026-05-18T03:10:14Z","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":"755ab1bcbd135b9f1d91abc18a3f23982ab58bab9e2c68986a62304a32cd7a60","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-11-06T16:07:26Z","title_canon_sha256":"403227be97bd85a69085b180348aff7c16ce81b624c88e989b919f2ae812c912"},"schema_version":"1.0","source":{"id":"1011.1567","kind":"arxiv","version":4}},"canonical_sha256":"df08abef20f7331db59411afbc357524a4be8e8241e272662489b51409cf3b7b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"df08abef20f7331db59411afbc357524a4be8e8241e272662489b51409cf3b7b","first_computed_at":"2026-05-18T03:10:14.796431Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:10:14.796431Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"U4qcVtAmDY5WeUiecBAQRG4R8E1NumFOjw7EsuT/dfma67j7TvSuLCaWjtQy+9SEzpVYUC2EjX3gkQezjuiPBw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:10:14.797138Z","signed_message":"canonical_sha256_bytes"},"source_id":"1011.1567","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ec85d8f7faa4926e4c5916b8d8b3226c32fef0ce20bd67f945d3078e7ae8c4c5","sha256:c409325cbe6c0259805b2c89ebbf380a8af9a887d96d24423846483ee0505252"],"state_sha256":"f1bdc9072df009e4247190cbf5a4e934d10f364b4a564f47bce1c5e048c65d3a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wA8kRh6Or+ii4fFQYHOfY7FJ4kpHcvh1xlrovlHTuOc/hkdc/VEdCSa98M1xbq7sdJyEHJSN4x8xAPozWyAbAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T11:58:11.216041Z","bundle_sha256":"474bb5c157832234e208de68fc4bff3d039a4bb3806b7fe8f8812e2c5125c42b"}}