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For large $\\lambda$, computing the partition function (namely, the normalizing constant which makes the weighting a probability distribution on a finite graph) on graphs of maximum degree $D\\ge 3$, is a well known computationally challenging problem. More concretely, let $\\lambda_c(T_D)$ denote the critical value for the so-called uniqueness threshold of the hard-core model on the infinite D-regular tree; rec"},"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":"1105.0914","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-05-04T19:52:13Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"e6495d3a0693b9392bbc9431c2461f8405148cb44681f057bc55b428c40a5735","abstract_canon_sha256":"f4aeafaee7ded3c9214f17657bd7388a62270941fe64cb8bb6725c4f78d40a66"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:15:36.665302Z","signature_b64":"U7ZoUT30LWXGHqZJ9jeqxR9GHdu4dMRsV+/TYm6Kf/RmjNjiG+NQ0MKRpwej0fFZlf2C9EqAB0suVojkq5jjDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"52f53c44050f06887d93c6d3bb8418d66cc1b2e63267634e6331f5f80bd727dd","last_reissued_at":"2026-05-18T04:15:36.664761Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:15:36.664761Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Improved Mixing Condition on the Grid for Counting and Sampling Independent Sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.PR","authors_text":"Eric Vigoda, Jinwoo Shin, Linji Yang, Prasad Tetali, Ricardo Restrepo","submitted_at":"2011-05-04T19:52:13Z","abstract_excerpt":"We study the hard-core model defined on independent sets, where each independent set I in a graph G is weighted proportionally to $\\lambda^{|I|}$, for a positive real parameter $\\lambda$. 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