Geodesic
Kac–Ward operators, Kasteleyn operators, and s -holomorphicity on arbitrary surface graphs
Annales de l’Institut Henri Poincaré D, Tome 2 (2015) no. 2, pp. 113-168

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�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 de�fine 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 de�fine 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.

Publié le :
DOI : 10.4171/aihpd/16
Classification : 82-XX, 57-XX
Keywords: Kac-Ward operator, Kasteleyn operator, s-holomorphic functions, Ising model 
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     title = {Kac{\textendash}Ward operators, {Kasteleyn} operators, and $s$-holomorphicity on arbitrary surface graphs},
     journal = {Annales de l{\textquoteright}Institut Henri Poincar\'e D},
     pages = {113--168},
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Cimasoni, David. Kac–Ward operators, Kasteleyn operators, and $s$-holomorphicity on arbitrary surface graphs. Annales de l’Institut Henri Poincaré D, Tome 2 (2015) no. 2, pp. 113-168. doi: 10.4171/aihpd/16

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