{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:YALAPC3IUBADDQYDLENWDAYQAP","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":"3b5204de3d0ec6aab06ce682cbf8ca90cf0c0098135fba1110f1b1b9d95d40b3","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-09-13T20:30:12Z","title_canon_sha256":"8442945d59b97908350028f1652b91137a2eef3334e89eaeee9be1e6e314befe"},"schema_version":"1.0","source":{"id":"1609.04040","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.04040","created_at":"2026-05-18T01:04:40Z"},{"alias_kind":"arxiv_version","alias_value":"1609.04040v1","created_at":"2026-05-18T01:04:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.04040","created_at":"2026-05-18T01:04:40Z"},{"alias_kind":"pith_short_12","alias_value":"YALAPC3IUBAD","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_16","alias_value":"YALAPC3IUBADDQYD","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_8","alias_value":"YALAPC3I","created_at":"2026-05-18T12:30:53Z"}],"graph_snapshots":[{"event_id":"sha256:14b69f8b70def683eba8e9d4323d18a8b8486e083208fab8d5964bd6cbe14afd","target":"graph","created_at":"2026-05-18T01:04:40Z","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":"Let $(G,\\rho)$ be a stationary random graph, and use $B^G_{\\rho}(r)$ to denote the ball of radius $r$ about $\\rho$ in $G$. Suppose that $(G,\\rho)$ has annealed polynomial growth, in the sense that $\\mathbb{E}[|B^G_{\\rho}(r)|] \\leq O(r^k)$ for some $k > 0$ and every $r \\geq 1$.\n  Then there is an infinite sequence of times $\\{t_n\\}$ at which the random walk $\\{X_t\\}$ on $(G,\\rho)$ is at most diffusive: Almost surely (over the choice of $(G,\\rho)$), there is a number $C > 0$ such that \\[ \\mathbb{E} \\left[\\mathrm{dist}_G(X_0, X_{t_n})^2 \\mid X_0 = \\rho, (G,\\rho)\\right]\\leq C t_n\\qquad \\forall n \\","authors_text":"James R. Lee, Shirshendu Ganguly, Yuval Peres","cross_cats":["math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-09-13T20:30:12Z","title":"Diffusive estimates for random walks on stationary random graphs of polynomial growth"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.04040","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":"sha256:44152ab56d9e098d2de2216522c333bc3da63d73b67f5a1096b64fecf4797705","target":"record","created_at":"2026-05-18T01:04:40Z","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":"3b5204de3d0ec6aab06ce682cbf8ca90cf0c0098135fba1110f1b1b9d95d40b3","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-09-13T20:30:12Z","title_canon_sha256":"8442945d59b97908350028f1652b91137a2eef3334e89eaeee9be1e6e314befe"},"schema_version":"1.0","source":{"id":"1609.04040","kind":"arxiv","version":1}},"canonical_sha256":"c016078b68a04031c303591b61831003e2886866a14d6c10bc6a911b6faa1c11","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c016078b68a04031c303591b61831003e2886866a14d6c10bc6a911b6faa1c11","first_computed_at":"2026-05-18T01:04:40.836968Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:04:40.836968Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3A1R5fOtgtbSNhdVhZhanbTL1WM8ruFt6xaE0ntW4fVdIdUbXDbXNFekpzL2d/NFhbmPd36JZB2i9fvdwPdRBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:04:40.837584Z","signed_message":"canonical_sha256_bytes"},"source_id":"1609.04040","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:44152ab56d9e098d2de2216522c333bc3da63d73b67f5a1096b64fecf4797705","sha256:14b69f8b70def683eba8e9d4323d18a8b8486e083208fab8d5964bd6cbe14afd"],"state_sha256":"11a930b82934a4102c4c2162fc7eb91e6c57c0ceae1537dcc18dac69cac40b10"}