{"paper":{"title":"Four-Cycle Free Graphs, Height Functions, the Pivot Property and Entropy Minimality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Nishant Chandgotia","submitted_at":"2014-11-14T20:02:04Z","abstract_excerpt":"Fix $d\\geq 2$. Given a finite undirected graph ${\\mathcal{H}}$ without self-loops and multiple edges, consider the corresponding `vertex' shift, $Hom(\\mathbb{Z}^d, \\mathcal{H})$ denoted by $X_{\\mathcal{H}}$. In this paper we focus on $\\mathcal{H}$ which is `four-cycle free'. The two main results of this paper are: $X_{\\mathcal{H}}$ has the pivot property, meaning that for all distinct configurations $x,y\\in X_{\\mathcal{H}}$ which differ only at finitely many sites there is a sequence of configurations $x=x^1, x^2, \\ldots, x^n=y\\in X_{{\\mathcal{H}}}$ for which the successive configurations $(x^"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.4029","kind":"arxiv","version":3},"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"}