{"paper":{"title":"The self-dual point of the two-dimensional random-cluster model is critical for $q\\geq 1$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Hugo Duminil-Copin, Vincent Beffara","submitted_at":"2010-06-25T21:37:56Z","abstract_excerpt":"We prove a long-standing conjecture on random-cluster models, namely that the critical point for such models with parameter $q\\geq1$ on the square lattice is equal to the self-dual point $p_{sd}(q) = \\sqrt q /(1+\\sqrt q)$. This gives a proof that the critical temperature of the $q$-state Potts model is equal to $\\log (1+\\sqrt q)$ for all $q\\geq 2$. We further prove that the transition is sharp, meaning that there is exponential decay of correlations in the sub-critical phase. The techniques of this paper are rigorous and valid for all $q\\geq 1$, in contrast to earlier methods valid only for ce"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1006.5073","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}