Absence of zeros implies strong spatial mixing

In this paper we show that absence of complex zeros of the partition function of the hard-core model on any family of bounded degree graphs that is closed under taking induced subgraphs implies that the associated probability measure, the hard-core measure, satisfies strong spatial mixing on that family. As a corollary we obtain that the hard-core measure on the family of bounded degree claw-free graphs satisfies strong spatial mixing for every value of the fugacity parameter. We furthermore derive strong spatial mixing for graph homomorphism measures from absence of zeros of the graph homomorphism partition function.