Phase transition free regions in the Ising model via the Kac-Ward operator

We investigate the spectral radius and operator norm of the Kac-Ward transition matrix for the Ising model on a general planar graph. We then use the obtained results to identify regions in the complex plane where the free energy density limits are analytic functions of the inverse temperature. The bound turns out to be optimal in the case of isoradial graphs, i.e. it yields criticality of the self-dual Z-invariant coupling constants.