2011/01/31 by David Cimasoni · 1 citation
Mathematics · Physics and Astronomy · #Graph theory and applications #Markov Chains and Monte Carlo Methods #Theoretical and Computational Physics #math-ph #math.GT #math.MP #msc:05C50 #msc:57M15 #msc:82B20
paper · pdf · doi:10.1007/s00220-012-1575-z
published as Comm. Math. Phys. 316 (2012), no. 1, 99-126 · 30 pages, 10 figures; added section 4.4 in version 3
arxiv created 2012/08/08 · openalex publication_date 2012/10/04 · arxiv updated 2015/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
The Kac-Ward formula allows to compute the Ising partition function on any finite graph G from the determinant of 22g matrices, where g is the genus of a surface in which G embeds. We show that in the case of isoradially embedded graphs with critical weights, these determinants have quite remarkable properties. First of all, they satisfy some generalized Kramers-Wannier duality: there is an explicit equality relating the determinants associated to a graph and to its dual graph. Also, they are proportional to the determinants of the discrete critical Laplacians on the graph G, exactly when the genus g is zero or one. Finally, they share several formal properties with the Ray-Singer ∂-torsions of the Riemann surface in which G embeds.