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From $\\Phi$ one can construct a sequence of corresponding models on the graphs $G_n$. Let $\\{\\mu_n\\}$ be the resulting Gibbs measures. Here we assume that $\\{\\mu_{n}\\}$ converges to some limiting Gibbs measure $\\mu$ on $T_{d}$ in the local weak$^*$ sense, and study the consequences of this convergence for the specific entropies $|V_n|^{-1}H(\\mu_n)$. We show that the limit"},"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":"1705.03589","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-05-10T02:29:02Z","cross_cats_sorted":[],"title_canon_sha256":"6cb5a2219e66f3a4b827e5c449be01cb422ad9af53c27a65185f2ed2c7c37a9b","abstract_canon_sha256":"3bad7587a05bf537c4dc4bca1a3cda6f5c25f41f6bbf1ce4fb3be6d1e6d88093"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:21:28.157386Z","signature_b64":"L6GZ2ddRA4dP3HcvXBllIST23VxNoNiN/bl5r3n55rBBQz8fdoYWXPX1Enu/MV/9syF52nvlpdYKF0aNWbzkBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9b81d482dd530a07523bf918d432e198255c57855687aaebacef33692e6bcbd7","last_reissued_at":"2026-05-18T00:21:28.156684Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:21:28.156684Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Gibbs measures over locally tree-like graphs and percolative entropy over infinite regular trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Moumanti Podder, Tim Austin","submitted_at":"2017-05-10T02:29:02Z","abstract_excerpt":"Consider a statistical physical model on the $d$-regular infinite tree $T_{d}$ described by a set of interactions $\\Phi$. 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