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We show that if $Q_1$ and $Q_2$ are two $\\Gamma$-equivariant positive contractions on $\\ell^2(\\Gamma)$ or on $\\ell^2(\\mathsf{E})$ with $Q_1 \\le Q_2$, then there exists a $\\Gamma$-invariant monotone coupling of the corresponding determinantal probability measures witnessing the stochastic domination ${\\bf P}^{Q_1} \\p"},"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":"1402.0969","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2014-02-05T08:34:11Z","cross_cats_sorted":["math.GR","math.OA"],"title_canon_sha256":"32fecda533ebc747c56e72b70b65b57dfc7d63046bd8f3d0796321c2504d27b0","abstract_canon_sha256":"7981d40052e1129355848a43ee1231bc20b6465004311b3f85fffe51ba83b389"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:53:35.086943Z","signature_b64":"3WD2iIvhA9mjCNeDt7F4DC3HGiLXS84JtoGX1TuHzq267s7vTd6eh8V2BG0mPN7pTWtUKteH/bAdTmJZax+YBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"301a1e6902a58e3a4be4393d502b0af3f1554de4b2aafe8fd158916c389822cc","last_reissued_at":"2026-05-17T23:53:35.086339Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:53:35.086339Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Invariant Coupling of Determinantal Measures on Sofic Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR","math.OA"],"primary_cat":"math.PR","authors_text":"Andreas Thom, Russell Lyons","submitted_at":"2014-02-05T08:34:11Z","abstract_excerpt":"To any positive contraction $Q$ on $\\ell^2(W)$, there is associated a determinantal probability measure ${\\mathbf P}^Q$ on $2^W$, where $W$ is a denumerable set. 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