{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:7NVAA4EXEJF2RYBA7GQQKNAP4G","short_pith_number":"pith:7NVAA4EX","canonical_record":{"source":{"id":"1602.05598","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-02-17T21:16:32Z","cross_cats_sorted":[],"title_canon_sha256":"97b0797dec3be0f49094dd3b3423b9825110247b628b5113fee378ed71d12a23","abstract_canon_sha256":"06e3e31bc1b32c026ec129668cb5b83eeaa7ad72523641ccd4655591015639fd"},"schema_version":"1.0"},"canonical_sha256":"fb6a007097224ba8e020f9a105340fe182ce1af34de7052ebd2d2ba15c9f4065","source":{"kind":"arxiv","id":"1602.05598","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.05598","created_at":"2026-05-18T00:31:56Z"},{"alias_kind":"arxiv_version","alias_value":"1602.05598v2","created_at":"2026-05-18T00:31:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.05598","created_at":"2026-05-18T00:31:56Z"},{"alias_kind":"pith_short_12","alias_value":"7NVAA4EXEJF2","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"7NVAA4EXEJF2RYBA","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"7NVAA4EX","created_at":"2026-05-18T12:30:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:7NVAA4EXEJF2RYBA7GQQKNAP4G","target":"record","payload":{"canonical_record":{"source":{"id":"1602.05598","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-02-17T21:16:32Z","cross_cats_sorted":[],"title_canon_sha256":"97b0797dec3be0f49094dd3b3423b9825110247b628b5113fee378ed71d12a23","abstract_canon_sha256":"06e3e31bc1b32c026ec129668cb5b83eeaa7ad72523641ccd4655591015639fd"},"schema_version":"1.0"},"canonical_sha256":"fb6a007097224ba8e020f9a105340fe182ce1af34de7052ebd2d2ba15c9f4065","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:31:56.248087Z","signature_b64":"cVUr4r6yJNwC9n+cvtoPLc6DXIyBDIbnmJshzc590j4ddcLC5fHSlzddQAI8Ub18rmVr5TVLtFknwX0j0dIvCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fb6a007097224ba8e020f9a105340fe182ce1af34de7052ebd2d2ba15c9f4065","last_reissued_at":"2026-05-18T00:31:56.247740Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:31:56.247740Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1602.05598","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":"2026-05-18T00:31:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Aub6nmqb7UvleIui8sR2zwGOc+6CF8asKAy6ut6DLR/n76+X4jnxn6Kefq7YrDQy7UxNb696fxwhsAuHxvZ+Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T12:38:44.147803Z"},"content_sha256":"01a6177f73b83dab17907ac794199aba1348f8342607b1615e4dc3334068f697","schema_version":"1.0","event_id":"sha256:01a6177f73b83dab17907ac794199aba1348f8342607b1615e4dc3334068f697"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:7NVAA4EXEJF2RYBA7GQQKNAP4G","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Isoperimetry in supercritical bond percolation in dimensions three and higher","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Julian Gold","submitted_at":"2016-02-17T21:16:32Z","abstract_excerpt":"We study the isoperimetric subgraphs of the infinite cluster $\\textbf{C}_\\infty$ for supercritical bond percolation on $\\mathbb{Z}^d$ with $d\\geq 3$. Specifically, we consider the subgraphs of $\\textbf{C}_\\infty \\cap [-n,n]^d$ which have minimal open edge boundary to volume ratio. We prove a shape theorem for these subgraphs, obtaining that when suitably rescaled, these subgraphs converge almost surely to a translate of a deterministic shape. This deterministic shape is itself an isoperimetric set for a norm we construct. As a corollary, we obtain sharp asymptotics on a natural modification of"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.05598","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":"2026-05-18T00:31:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MluaNrY/NZzJjufpXBE3dyEPhHgmhFLgGdtVhtjmhNjTGg/PxIWT4sSdF6fUrXFwyHrPsx+o/NR4FYaURV/wCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T12:38:44.148270Z"},"content_sha256":"20944d4c19f1914e5c22ab973171f6ae2fc54cc8dd685fd4b64871dbf8f1a621","schema_version":"1.0","event_id":"sha256:20944d4c19f1914e5c22ab973171f6ae2fc54cc8dd685fd4b64871dbf8f1a621"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7NVAA4EXEJF2RYBA7GQQKNAP4G/bundle.json","state_url":"https://pith.science/pith/7NVAA4EXEJF2RYBA7GQQKNAP4G/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7NVAA4EXEJF2RYBA7GQQKNAP4G/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":"2026-06-23T12:38:44Z","links":{"resolver":"https://pith.science/pith/7NVAA4EXEJF2RYBA7GQQKNAP4G","bundle":"https://pith.science/pith/7NVAA4EXEJF2RYBA7GQQKNAP4G/bundle.json","state":"https://pith.science/pith/7NVAA4EXEJF2RYBA7GQQKNAP4G/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7NVAA4EXEJF2RYBA7GQQKNAP4G/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:7NVAA4EXEJF2RYBA7GQQKNAP4G","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":"06e3e31bc1b32c026ec129668cb5b83eeaa7ad72523641ccd4655591015639fd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-02-17T21:16:32Z","title_canon_sha256":"97b0797dec3be0f49094dd3b3423b9825110247b628b5113fee378ed71d12a23"},"schema_version":"1.0","source":{"id":"1602.05598","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.05598","created_at":"2026-05-18T00:31:56Z"},{"alias_kind":"arxiv_version","alias_value":"1602.05598v2","created_at":"2026-05-18T00:31:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.05598","created_at":"2026-05-18T00:31:56Z"},{"alias_kind":"pith_short_12","alias_value":"7NVAA4EXEJF2","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"7NVAA4EXEJF2RYBA","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"7NVAA4EX","created_at":"2026-05-18T12:30:04Z"}],"graph_snapshots":[{"event_id":"sha256:20944d4c19f1914e5c22ab973171f6ae2fc54cc8dd685fd4b64871dbf8f1a621","target":"graph","created_at":"2026-05-18T00:31:56Z","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 isoperimetric subgraphs of the infinite cluster $\\textbf{C}_\\infty$ for supercritical bond percolation on $\\mathbb{Z}^d$ with $d\\geq 3$. Specifically, we consider the subgraphs of $\\textbf{C}_\\infty \\cap [-n,n]^d$ which have minimal open edge boundary to volume ratio. We prove a shape theorem for these subgraphs, obtaining that when suitably rescaled, these subgraphs converge almost surely to a translate of a deterministic shape. This deterministic shape is itself an isoperimetric set for a norm we construct. As a corollary, we obtain sharp asymptotics on a natural modification of","authors_text":"Julian Gold","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-02-17T21:16:32Z","title":"Isoperimetry in supercritical bond percolation in dimensions three and higher"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.05598","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":"sha256:01a6177f73b83dab17907ac794199aba1348f8342607b1615e4dc3334068f697","target":"record","created_at":"2026-05-18T00:31:56Z","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":"06e3e31bc1b32c026ec129668cb5b83eeaa7ad72523641ccd4655591015639fd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-02-17T21:16:32Z","title_canon_sha256":"97b0797dec3be0f49094dd3b3423b9825110247b628b5113fee378ed71d12a23"},"schema_version":"1.0","source":{"id":"1602.05598","kind":"arxiv","version":2}},"canonical_sha256":"fb6a007097224ba8e020f9a105340fe182ce1af34de7052ebd2d2ba15c9f4065","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fb6a007097224ba8e020f9a105340fe182ce1af34de7052ebd2d2ba15c9f4065","first_computed_at":"2026-05-18T00:31:56.247740Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:31:56.247740Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cVUr4r6yJNwC9n+cvtoPLc6DXIyBDIbnmJshzc590j4ddcLC5fHSlzddQAI8Ub18rmVr5TVLtFknwX0j0dIvCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:31:56.248087Z","signed_message":"canonical_sha256_bytes"},"source_id":"1602.05598","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:01a6177f73b83dab17907ac794199aba1348f8342607b1615e4dc3334068f697","sha256:20944d4c19f1914e5c22ab973171f6ae2fc54cc8dd685fd4b64871dbf8f1a621"],"state_sha256":"2881ae8501bb45ce60e6e8301ddbe0622a75e8c9f0da96b1df81f552f93734f6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9Y3z1ssUBm7J+yd75GlvVjxci1swQ4D9/oK+nLVOqwDQl4INOc0yGxqOdnTL9pu5Hh/dsc3HSbqqLetveAQDCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T12:38:44.150794Z","bundle_sha256":"0783b3b3dfb5ae4c773974b62d2e83da89acb6714a970aa2d719fce8cad68759"}}