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Annales de l’Institut Henri Poincaré D


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Volume 2, Issue 2, 2015, pp. 113–168
DOI: 10.4171/AIHPD/16

Published online: 2015-05-29

Kac–Ward operators, Kasteleyn operators, and s-holomorphicity on arbitrary surface graphs

David Cimasoni[1]

(1) Université de Genève, Switzerland

The conformal invariance and universality results of Chelkak-Smirnov on the two-dimensional Ising model hold for isoradial planar graphs with critical weights. Motivated by the problem of extending these results to a wider class of graphs, we define a generalized notion of s-holomorphicity for functions on arbitrary weighted surface graphs. We then give three criteria for s-holomorphicity involving the Kac–Ward, Kasteleyn, and discrete Dirac operators, respectively. Also, we show that some crucial results known to hold in the planar isoradial case extend to this general setting: in particular, spin-Ising fermionic observables are s-holomorphic, and it is possible to define a discrete version of the integral of the square of an s-holomorphic function. Along the way, we obtain a duality result for Kac–Ward determinants on arbitrary weighted surface graphs.

Keywords: Kac-Ward operator, Kasteleyn operator, s-holomorphic functions, Ising model

Cimasoni David: Kac–Ward operators, Kasteleyn operators, and s-holomorphicity on arbitrary surface graphs. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 2 (2015), 113-168. doi: 10.4171/AIHPD/16