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For specific choices of the parameters $a, b$ the two-valued sets have the law of the CLE$_4$ carpet, the law of the union of level lines between all pairs of boundary points, or, conjecturally, the law of the interfaces of the scaling limit of XOR-Ising model.\n  Two-valued sets "},"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":"1801.03828","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2018-01-11T15:46:22Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"d251353e3bcc8892aeaeafa4f3e6b12de1f5f062eaba79870a0f0b064741c652","abstract_canon_sha256":"9500a51e4759230ba27106fc791b5befe080b3718c539f8331f88875f7c75974"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:17:10.954468Z","signature_b64":"SXjnWFdjdVYJG1AfeUjHLgah3eXy3xjFVdwFasNayEbK7I4tUNHFpseaz2IpsBBjx/87wSbgWNDUWlB2grPCAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"180ef2e5e0fe071f704ca7f827aacdfcffb421c7e1f1a81f71392ea07dcde4f8","last_reissued_at":"2026-05-18T00:17:10.953823Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:17:10.953823Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Two-valued local sets of the 2D continuum Gaussian free field: connectivity, labels, and induced metrics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Avelio Sep\\'ulveda, Juhan Aru","submitted_at":"2018-01-11T15:46:22Z","abstract_excerpt":"We study two-valued local sets, $\\mathbb{A}_{-a,b}$, of the two-dimensional continuum Gaussian free field (GFF) with zero boundary condition in simply connected domains. 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