{"paper":{"title":"The Laplacian and $\\bar\\partial$ operators on critical planar graphs","license":"","headline":"","cross_cats":["math.MP","math.SP"],"primary_cat":"math-ph","authors_text":"Richard Kenyon","submitted_at":"2002-02-13T15:47:29Z","abstract_excerpt":"On a periodic planar graph whose edge weights satisfy a certain simple geometric condition, the discrete Laplacian and d-bar operators have the property that their determinants and inverses only depend on the local geometry of the graph.\n  We obtain explicit expressions for the logarithms of the (normalized) determinants, as well as the inverses of these operators.\n  We relate the logarithm of the determinants to the volume plus mean curvature of an associated hyperbolic ideal polyhedron.\n  In the associated dimer and spanning tree models, for which the determinants of d-bar and the Laplacian "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math-ph/0202018","kind":"arxiv","version":1},"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"}