{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:LW6CIK6XQYV7NX3OQWXX2TVMWY","short_pith_number":"pith:LW6CIK6X","canonical_record":{"source":{"id":"1512.02790","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-12-09T08:53:29Z","cross_cats_sorted":[],"title_canon_sha256":"80cb02ee52db7d004f340284ae01e13d81b26005f162819f62a619ce279641ba","abstract_canon_sha256":"140ea49bf4ea68986a2285435edc93d00989c6cda35b4ef54faa585ad5cac6f8"},"schema_version":"1.0"},"canonical_sha256":"5dbc242bd7862bf6df6e85af7d4eacb633299ccc52e7a92ce2c44811edae3926","source":{"kind":"arxiv","id":"1512.02790","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.02790","created_at":"2026-05-18T01:24:43Z"},{"alias_kind":"arxiv_version","alias_value":"1512.02790v1","created_at":"2026-05-18T01:24:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.02790","created_at":"2026-05-18T01:24:43Z"},{"alias_kind":"pith_short_12","alias_value":"LW6CIK6XQYV7","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_16","alias_value":"LW6CIK6XQYV7NX3O","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_8","alias_value":"LW6CIK6X","created_at":"2026-05-18T12:29:29Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:LW6CIK6XQYV7NX3OQWXX2TVMWY","target":"record","payload":{"canonical_record":{"source":{"id":"1512.02790","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-12-09T08:53:29Z","cross_cats_sorted":[],"title_canon_sha256":"80cb02ee52db7d004f340284ae01e13d81b26005f162819f62a619ce279641ba","abstract_canon_sha256":"140ea49bf4ea68986a2285435edc93d00989c6cda35b4ef54faa585ad5cac6f8"},"schema_version":"1.0"},"canonical_sha256":"5dbc242bd7862bf6df6e85af7d4eacb633299ccc52e7a92ce2c44811edae3926","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:24:43.142127Z","signature_b64":"XW/ohEDh58YEDOxLN8bdZL5/JJJ9PXBmxcoGpNU9O5Vr1IYHIcs1ny0gczmRcoQiNwliLKK+isgF/7qi/h1WBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5dbc242bd7862bf6df6e85af7d4eacb633299ccc52e7a92ce2c44811edae3926","last_reissued_at":"2026-05-18T01:24:43.141740Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:24:43.141740Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1512.02790","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:24:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"q9Bv7msIqUWwC4QpPBvorNXDYPEFNCFJKJrhb8/0Xff4JMCIFfN8nvalR/Am+oLfaNKsToln0fwOcmv3ScDhDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T14:54:25.415717Z"},"content_sha256":"cf94c28ab81b2d580756fc9a6b355e52c7bdb7e3a3174696d64dec0e11dc9654","schema_version":"1.0","event_id":"sha256:cf94c28ab81b2d580756fc9a6b355e52c7bdb7e3a3174696d64dec0e11dc9654"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:LW6CIK6XQYV7NX3OQWXX2TVMWY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Mixing time for the random walk on the range of the random walk on tori","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Artem Sapozhnikov, Ji\\v{r}\\'i \\v{C}ern\\'y","submitted_at":"2015-12-09T08:53:29Z","abstract_excerpt":"Consider the subgraph of the discrete $d$-dimensional torus of size length $N$, $d\\ge3$, induced by the range of the simple random walk on the torus run until the time $uN^d$. We prove that for all $d\\ge 3$ and $u>0$, the mixing time for the random walk on this subgraph is of order $N^2$ with probability at least $1 - Ce^{-(\\log N)^2}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.02790","kind":"arxiv","version":1},"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":"2026-05-18T01:24:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WZX0QL++vJLnkehQtTAjz/6tt3kfeAEMT6YoGJJeavJSXle//yKBK19Tl/qnTwzy+p/Lnog54uzHW1z/iu2OCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-21T14:54:25.416063Z"},"content_sha256":"d4472715aa9011c26901ee585b11a48cbec87a46a6440a8e50fd41fc4d11b5ab","schema_version":"1.0","event_id":"sha256:d4472715aa9011c26901ee585b11a48cbec87a46a6440a8e50fd41fc4d11b5ab"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LW6CIK6XQYV7NX3OQWXX2TVMWY/bundle.json","state_url":"https://pith.science/pith/LW6CIK6XQYV7NX3OQWXX2TVMWY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LW6CIK6XQYV7NX3OQWXX2TVMWY/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":"2026-06-21T14:54:25Z","links":{"resolver":"https://pith.science/pith/LW6CIK6XQYV7NX3OQWXX2TVMWY","bundle":"https://pith.science/pith/LW6CIK6XQYV7NX3OQWXX2TVMWY/bundle.json","state":"https://pith.science/pith/LW6CIK6XQYV7NX3OQWXX2TVMWY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LW6CIK6XQYV7NX3OQWXX2TVMWY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:LW6CIK6XQYV7NX3OQWXX2TVMWY","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":"140ea49bf4ea68986a2285435edc93d00989c6cda35b4ef54faa585ad5cac6f8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-12-09T08:53:29Z","title_canon_sha256":"80cb02ee52db7d004f340284ae01e13d81b26005f162819f62a619ce279641ba"},"schema_version":"1.0","source":{"id":"1512.02790","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1512.02790","created_at":"2026-05-18T01:24:43Z"},{"alias_kind":"arxiv_version","alias_value":"1512.02790v1","created_at":"2026-05-18T01:24:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.02790","created_at":"2026-05-18T01:24:43Z"},{"alias_kind":"pith_short_12","alias_value":"LW6CIK6XQYV7","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_16","alias_value":"LW6CIK6XQYV7NX3O","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_8","alias_value":"LW6CIK6X","created_at":"2026-05-18T12:29:29Z"}],"graph_snapshots":[{"event_id":"sha256:d4472715aa9011c26901ee585b11a48cbec87a46a6440a8e50fd41fc4d11b5ab","target":"graph","created_at":"2026-05-18T01:24:43Z","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":"Consider the subgraph of the discrete $d$-dimensional torus of size length $N$, $d\\ge3$, induced by the range of the simple random walk on the torus run until the time $uN^d$. We prove that for all $d\\ge 3$ and $u>0$, the mixing time for the random walk on this subgraph is of order $N^2$ with probability at least $1 - Ce^{-(\\log N)^2}$.","authors_text":"Artem Sapozhnikov, Ji\\v{r}\\'i \\v{C}ern\\'y","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-12-09T08:53:29Z","title":"Mixing time for the random walk on the range of the random walk on tori"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.02790","kind":"arxiv","version":1},"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:cf94c28ab81b2d580756fc9a6b355e52c7bdb7e3a3174696d64dec0e11dc9654","target":"record","created_at":"2026-05-18T01:24:43Z","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":"140ea49bf4ea68986a2285435edc93d00989c6cda35b4ef54faa585ad5cac6f8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-12-09T08:53:29Z","title_canon_sha256":"80cb02ee52db7d004f340284ae01e13d81b26005f162819f62a619ce279641ba"},"schema_version":"1.0","source":{"id":"1512.02790","kind":"arxiv","version":1}},"canonical_sha256":"5dbc242bd7862bf6df6e85af7d4eacb633299ccc52e7a92ce2c44811edae3926","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5dbc242bd7862bf6df6e85af7d4eacb633299ccc52e7a92ce2c44811edae3926","first_computed_at":"2026-05-18T01:24:43.141740Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:24:43.141740Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XW/ohEDh58YEDOxLN8bdZL5/JJJ9PXBmxcoGpNU9O5Vr1IYHIcs1ny0gczmRcoQiNwliLKK+isgF/7qi/h1WBw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:24:43.142127Z","signed_message":"canonical_sha256_bytes"},"source_id":"1512.02790","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cf94c28ab81b2d580756fc9a6b355e52c7bdb7e3a3174696d64dec0e11dc9654","sha256:d4472715aa9011c26901ee585b11a48cbec87a46a6440a8e50fd41fc4d11b5ab"],"state_sha256":"a41c0768e65621327e68a53d66a3529cb42267a895dd07c8d3378e1aa76e7bf8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MtQihwLvRavN5RqZL+X3KH94wA1kBOefWz4vcZ0kvrWt1xPIruxYadjSvyx+JBkKVqfWDXev/Ug90s0UnPFdBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-21T14:54:25.418068Z","bundle_sha256":"b2e8dfbeed1bd9dbde8a44432964246d92b1a3b2ebc70f96470675f5329bdb2f"}}