{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:17:38Z","timestamp":1760145458683,"version":"build-2065373602"},"reference-count":38,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2024,7,27]],"date-time":"2024-07-27T00:00:00Z","timestamp":1722038400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We present a new combinatorial approach to the Ising model incorporating arbitrary bond weights on planar graphs. In contrast to existing methodologies, the exact free energy is expressed as the determinant of a set of ordered and disordered operators defined on a planar graph and the corresponding dual graph, respectively, thereby explicitly demonstrating the Kramers\u2013Wannier duality. The implications of our derived formula for the Random-Bond Ising Model are further elucidated.<\/jats:p>","DOI":"10.3390\/e26080636","type":"journal-article","created":{"date-parts":[[2024,7,29]],"date-time":"2024-07-29T12:27:43Z","timestamp":1722256063000},"page":"636","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Kramers\u2013Wannier Duality and Random-Bond Ising Model"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3048-7046","authenticated-orcid":false,"given":"Chaoming","family":"Song","sequence":"first","affiliation":[{"name":"Department of Physics, University of Miami, Coral Gables, FL 33146, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,7,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"252","DOI":"10.1103\/PhysRev.60.252","article-title":"Statistics of the two-dimensional ferromagnet. 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