Decay of correlations and zeros for the hard-core model

In a recent paper the last author proved that absence of complex zeros of the partition function of the hard-core model near a parameter $\lambda>0$ implies a form of correlation decay called strong spacial mixing. In this paper we investigate the reverse implication. We introduce a strengthening of strong spatial mixing that we call very strong spatial mixing (VSSM). Our main result is that if VSSM holds at a parameter $\lambda>0$ for a family of graphs, this implies that the partition function has no zeros near that parameter for each graph in the family. We also demonstrate that a closely related variant of very strong spatial mixing does not imply zero-freeness. As a consequence of our main result, we moreover obtain that VSSM implies spectral independence. Our proof relies on transforming the problem to the analysis of an induced non-autonomous dynamical system given by M\"obius transformations.