�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.
DOI : 10.4171/aihpd/16
Mots-clés : Kac-Ward operator, Kasteleyn operator, s-holomorphic functions, Ising model
@article{AIHPD_2015__2_2_113_0, author = {Cimasoni, David}, 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}, publisher = {EMS Press}, volume = {2}, number = {2}, year = {2015}, doi = {10.4171/aihpd/16}, zbl = {1334.82008}, language = {en}, url = {http://archive.numdam.org/articles/10.4171/aihpd/16/} }
TY - JOUR AU - Cimasoni, David TI - Kac–Ward operators, Kasteleyn operators, and $s$-holomorphicity on arbitrary surface graphs JO - Annales de l’Institut Henri Poincaré D PY - 2015 SP - 113 EP - 168 VL - 2 IS - 2 PB - EMS Press UR - http://archive.numdam.org/articles/10.4171/aihpd/16/ DO - 10.4171/aihpd/16 LA - en ID - AIHPD_2015__2_2_113_0 ER -
%0 Journal Article %A Cimasoni, David %T Kac–Ward operators, Kasteleyn operators, and $s$-holomorphicity on arbitrary surface graphs %J Annales de l’Institut Henri Poincaré D %D 2015 %P 113-168 %V 2 %N 2 %I EMS Press %U http://archive.numdam.org/articles/10.4171/aihpd/16/ %R 10.4171/aihpd/16 %G en %F AIHPD_2015__2_2_113_0
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. http://archive.numdam.org/articles/10.4171/aihpd/16/
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