The critical Ising model via Kac-Ward matrices

Details The critical Ising model via Kac-Ward matrices The critical Ising model via Kac-Ward matrices

2011-01-28

arxiv.org/abs/1101.5559v2

Physics

Mathematical Physics

28 pages, 10 figures; minor changes in version 2

Scientific paper

The Kac-Ward formula allows to compute the Ising partition function on any finite graph G from the determinant of 2^{2g} 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 admit a relatively simple combinatorial interpretation. Also, they satisfy some generalized Kramers-Wannier duality: there is an explicit equality relating the determinants associated to a graph and to its dual graph. Finally, they are proportional to the determinants of the discrete critical Laplacians on the graph G, exactly when the genus g is zero or one.

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