Decay of correlations and zeros for the hard-core model

Abstract

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 Möbius transformations.

28 pages; 3 figures; We removed two questions in the final section (as we found (negative) answers to them). The main body of the paper and in particular its main result(s) remain unchanged. We have also reordered the author alphabetically (which was for some reason not done in the first version)