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The result concerning percolation recovers a result of Kozma (2013), while the result concerning the Ising model is new.\n  We also present a new proof, using similar techniques, of a lemma of Schramm concerning the decay of the critical two-point function al"},"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":"1712.04911","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-12-13T18:30:51Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"4242939b8606a581854247c8cd8f635f865de622c45cd950c8a74d727008bbd6","abstract_canon_sha256":"3675dac864f2765613115dc9816f0973ce4204c93e92809788db14812580abb3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:56:35.166747Z","signature_b64":"iy5c1jXbhRAEA/y7cyxpYvIY331heVj/kcP1XXgdQQJ7XUIZ/JKFxu2n7VxbypVq6a8ep0ZRRXTUi6JEirdcBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bfae3ea6ca592bb22918f7efb4b1624b4bba334482f796e578b9f4c15d69dde1","last_reissued_at":"2026-05-17T23:56:35.166151Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:56:35.166151Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Statistical physics on a product of trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Tom Hutchcroft","submitted_at":"2017-12-13T18:30:51Z","abstract_excerpt":"Let $G$ be the product of finitely many trees $T_1\\times T_2 \\times \\cdots \\times T_N$, each of which is regular with degree at least three. 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