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We investigate the spectral radius and operator norm of the Kac-Wardtransition matrix for the Ising model on a general planar graph. We then usethe obtained results to identify regions in the complex plane where the freeenergy density limits are analytic functions of the inverse temperature. Thebound turns out to be optimal in the case of isoradial graphs, i.e. it yieldscriticality of the self-dual Z-invariant coupling constants.

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