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Assume $M$ has a real second-largest (in absolute value) eigenvalue $\\lambda$ with corresponding real eigenvector $\\nu \\neq 0$. Letting $\\sigma_v = \\nu_{\\xi_v}$, we consider the following root-state estimator, which was introduced by Mossel and Peres (2003) in the context of the \"recontruction problem\" on trees: \\begin{equation*} S_n = (b\\lambda)^{-n} \\s"},"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":"0908.2056","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-08-14T13:05:32Z","cross_cats_sorted":["cs.DS","math.ST","q-bio.PE","stat.TH"],"title_canon_sha256":"46ae3ecd5fdc7a67494d7a2ec2869eefa0772c381465323692522ae267efd2c3","abstract_canon_sha256":"41e4f6f8f069d26a6c99c072d2d007e70c648c13107fc5d83b8c01e31f2a981b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:14:06.042966Z","signature_b64":"7fQRCVAev8WodKyO8o55sfq5SD8UcDXE+gwv/pOks1k6EBYJ9X/rv3Z0ZlPAr4oXR0+qSnCLlhc7EscXrvK1DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1e1891b1986f455e475a765e901f1d0c2383ca84c8fa0a460986cee59d781371","last_reissued_at":"2026-05-18T04:14:06.042167Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:14:06.042167Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Reconstruction on Trees: Exponential Moment Bounds for Linear Estimators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS","math.ST","q-bio.PE","stat.TH"],"primary_cat":"math.PR","authors_text":"Sebastien Roch, Yuval Peres","submitted_at":"2009-08-14T13:05:32Z","abstract_excerpt":"Consider a Markov chain $(\\xi_v)_{v \\in V} \\in [k]^V$ on the infinite $b$-ary tree $T = (V,E)$ with irreducible edge transition matrix $M$, where $b \\geq 2$, $k \\geq 2$ and $[k] = \\{1,...,k\\}$. We denote by $L_n$ the level-$n$ vertices of $T$. Assume $M$ has a real second-largest (in absolute value) eigenvalue $\\lambda$ with corresponding real eigenvector $\\nu \\neq 0$. Letting $\\sigma_v = \\nu_{\\xi_v}$, we consider the following root-state estimator, which was introduced by Mossel and Peres (2003) in the context of the \"recontruction problem\" on trees: \\begin{equation*} S_n = (b\\lambda)^{-n} \\s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0908.2056","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"0908.2056","created_at":"2026-05-18T04:14:06.042271+00:00"},{"alias_kind":"arxiv_version","alias_value":"0908.2056v1","created_at":"2026-05-18T04:14:06.042271+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0908.2056","created_at":"2026-05-18T04:14:06.042271+00:00"},{"alias_kind":"pith_short_12","alias_value":"DYMJDMMYN5CV","created_at":"2026-05-18T12:25:59.703012+00:00"},{"alias_kind":"pith_short_16","alias_value":"DYMJDMMYN5CV4R22","created_at":"2026-05-18T12:25:59.703012+00:00"},{"alias_kind":"pith_short_8","alias_value":"DYMJDMMY","created_at":"2026-05-18T12:25:59.703012+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/DYMJDMMYN5CV4R22OZPJAHY5BQ","json":"https://pith.science/pith/DYMJDMMYN5CV4R22OZPJAHY5BQ.json","graph_json":"https://pith.science/api/pith-number/DYMJDMMYN5CV4R22OZPJAHY5BQ/graph.json","events_json":"https://pith.science/api/pith-number/DYMJDMMYN5CV4R22OZPJAHY5BQ/events.json","paper":"https://pith.science/paper/DYMJDMMY"},"agent_actions":{"view_html":"https://pith.science/pith/DYMJDMMYN5CV4R22OZPJAHY5BQ","download_json":"https://pith.science/pith/DYMJDMMYN5CV4R22OZPJAHY5BQ.json","view_paper":"https://pith.science/paper/DYMJDMMY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0908.2056&json=true","fetch_graph":"https://pith.science/api/pith-number/DYMJDMMYN5CV4R22OZPJAHY5BQ/graph.json","fetch_events":"https://pith.science/api/pith-number/DYMJDMMYN5CV4R22OZPJAHY5BQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DYMJDMMYN5CV4R22OZPJAHY5BQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DYMJDMMYN5CV4R22OZPJAHY5BQ/action/storage_attestation","attest_author":"https://pith.science/pith/DYMJDMMYN5CV4R22OZPJAHY5BQ/action/author_attestation","sign_citation":"https://pith.science/pith/DYMJDMMYN5CV4R22OZPJAHY5BQ/action/citation_signature","submit_replication":"https://pith.science/pith/DYMJDMMYN5CV4R22OZPJAHY5BQ/action/replication_record"}},"created_at":"2026-05-18T04:14:06.042271+00:00","updated_at":"2026-05-18T04:14:06.042271+00:00"}