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The lamplighter chain $X^\\diamond$ associated with $X$ is the random walk on the wreath product $\\CG^\\diamond = \\Z_2 \\wr \\CG$, the graph whose vertices consist of pairs $(f,x)$ where $f$ is a labeling of the vertices of $\\CG$ by elements of $\\Z_2$ and $x$ is a vertex in $\\CG$. There is an edge between $(f,x)$ and $(g,y)$ in $\\CG^\\diamond$ if and only if $x$ is adjacent to $y$ in $\\CG$ and $f(z) = g(z)$ for all $z \\neq x,y$. In each step, $X^\\diamond$ moves from a configuration $(f,x)$ by updating $x$ to $y$"},"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":"1109.4281","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-09-20T12:31:40Z","cross_cats_sorted":[],"title_canon_sha256":"6574d36865c59c11346f787cd16a75a858f3815c58bf0dc9fce94a5e4ecfba0e","abstract_canon_sha256":"e7d16f6244251d9682e650e72496ca74b395cd225c843cb232602f9aaa1fa0a5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:59:35.099257Z","signature_b64":"T2QwTnxwDtGTSMKB2JlUC5axH/6ahh+B5oYyGjdyARmtWvletcVOKvO6sMEG2GvN3DUduJPzgWFBsPWn5nYYAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"62609877387e874e738044bc5a7ffbb93e5004c6f1642106234d86e11574ecfd","last_reissued_at":"2026-05-18T00:59:35.098587Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:59:35.098587Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Uniform mixing time for Random Walk on Lamplighter Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Jason Miller, J\\'ulia Komj\\'athy, Yuval Peres","submitted_at":"2011-09-20T12:31:40Z","abstract_excerpt":"Suppose that $\\CG$ is a finite, connected graph and $X$ is a lazy random walk on $\\CG$. The lamplighter chain $X^\\diamond$ associated with $X$ is the random walk on the wreath product $\\CG^\\diamond = \\Z_2 \\wr \\CG$, the graph whose vertices consist of pairs $(f,x)$ where $f$ is a labeling of the vertices of $\\CG$ by elements of $\\Z_2$ and $x$ is a vertex in $\\CG$. There is an edge between $(f,x)$ and $(g,y)$ in $\\CG^\\diamond$ if and only if $x$ is adjacent to $y$ in $\\CG$ and $f(z) = g(z)$ for all $z \\neq x,y$. 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