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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 \\"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1609.04040","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-09-13T20:30:12Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"8442945d59b97908350028f1652b91137a2eef3334e89eaeee9be1e6e314befe","abstract_canon_sha256":"3b5204de3d0ec6aab06ce682cbf8ca90cf0c0098135fba1110f1b1b9d95d40b3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:40.837584Z","signature_b64":"3A1R5fOtgtbSNhdVhZhanbTL1WM8ruFt6xaE0ntW4fVdIdUbXDbXNFekpzL2d/NFhbmPd36JZB2i9fvdwPdRBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c016078b68a04031c303591b61831003e2886866a14d6c10bc6a911b6faa1c11","last_reissued_at":"2026-05-18T01:04:40.836968Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:40.836968Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Diffusive estimates for random walks on stationary random graphs of polynomial growth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.PR","authors_text":"James R. 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