{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:B7HH7IELCJNK6EJBNR4DENR42V","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":"c706eae6d7130e20a955340246cb26d51a2566637198aad079d578f08a607871","cross_cats_sorted":["math.PR"],"license":"","primary_cat":"math.FA","submitted_at":"2007-12-19T10:00:31Z","title_canon_sha256":"f2ec5de4fc84637a85c4d672487982ea2173efcbebdf9a44b7f6bc7962c9b16f"},"schema_version":"1.0","source":{"id":"0712.3135","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0712.3135","created_at":"2026-05-18T03:39:14Z"},{"alias_kind":"arxiv_version","alias_value":"0712.3135v2","created_at":"2026-05-18T03:39:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0712.3135","created_at":"2026-05-18T03:39:14Z"},{"alias_kind":"pith_short_12","alias_value":"B7HH7IELCJNK","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_16","alias_value":"B7HH7IELCJNK6EJB","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_8","alias_value":"B7HH7IEL","created_at":"2026-05-18T12:25:55Z"}],"graph_snapshots":[{"event_id":"sha256:30affb3f05c8c60da099acff9aac5b9e0bcd7d8dc7a88f0c90d94823599f624b","target":"graph","created_at":"2026-05-18T03:39:14Z","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 $G$ be a finitely generated group and $X$ its Cayley graph with respect to a finite, symmetric generating set $S$. Furthermore, let $H$ be a finite group and $H \\wr G$ the lamplighter group (wreath product) over $G$ with group of \"lamps\" $H$. We show that the spectral measure (Plancherel measure) of any symmetric \"switch--walk--switch\" random walk on $H \\wr G$ coincides with the expected spectral measure (integrated density of states) of the random walk with absorbing boundary on the cluster of the group identity for Bernoulli site percolation on $X$ with parameter $p = 1/|H|$. The return ","authors_text":"Franz Lehner, Markus Neuhauser, Wolfgang Woess","cross_cats":["math.PR"],"headline":"","license":"","primary_cat":"math.FA","submitted_at":"2007-12-19T10:00:31Z","title":"On the spectrum of lamplighter groups and percolation clusters"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0712.3135","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":"sha256:108c4a5fdf0898fc8e25df224e6d34e1abb962fa03c5534bc7ed49aedc6dec60","target":"record","created_at":"2026-05-18T03:39:14Z","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":"c706eae6d7130e20a955340246cb26d51a2566637198aad079d578f08a607871","cross_cats_sorted":["math.PR"],"license":"","primary_cat":"math.FA","submitted_at":"2007-12-19T10:00:31Z","title_canon_sha256":"f2ec5de4fc84637a85c4d672487982ea2173efcbebdf9a44b7f6bc7962c9b16f"},"schema_version":"1.0","source":{"id":"0712.3135","kind":"arxiv","version":2}},"canonical_sha256":"0fce7fa08b125aaf11216c7832363cd57130ac15acc29e00cfcbc39744b9e48e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0fce7fa08b125aaf11216c7832363cd57130ac15acc29e00cfcbc39744b9e48e","first_computed_at":"2026-05-18T03:39:14.943203Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:39:14.943203Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"USouUSg5UEVCjmUC8f8dtdJWVRt9clnewSCxGUnprgX41CorA6y0Dma39XJxWBdzmizC/Zdpl3h9F7dW+wmhCw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:39:14.943704Z","signed_message":"canonical_sha256_bytes"},"source_id":"0712.3135","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:108c4a5fdf0898fc8e25df224e6d34e1abb962fa03c5534bc7ed49aedc6dec60","sha256:30affb3f05c8c60da099acff9aac5b9e0bcd7d8dc7a88f0c90d94823599f624b"],"state_sha256":"3aa446dc25bf1a0c1c29f073e49ad3664ed59df1569c55cc1bde9710e57c113f"}