The fermionic observable in the Ising model and the inverse Kac-Ward operator

We show that the critical Kac-Ward operator on isoradial graphs acts in a certain sense as the operator of s-holomorphicity, and we identify the fermionic observable for the spin Ising model as the inverse of this operator. This result is partially a consequence of a more general observation that the inverse Kac-Ward operator for any planar graph is given by what we call a fermionic generating function. Furthermore, using bounds for the spectral radius and operator norm of the Kac-Ward transition matrix, we provide a general picture of the non-backtracking walk representation of the critical and supercritical inverse Kac-Ward operators on isoradial graphs.