Absence of zeros implies strong spatial mixing

TL;DR

This paper demonstrates that the absence of zeros in the partition function of certain graph models guarantees strong spatial mixing, with implications for hard-core and graph homomorphism measures.

Contribution

It establishes a zero-free region criterion that implies strong spatial mixing for the hard-core model and graph homomorphism measures on bounded degree graphs.

Findings

Absence of zeros implies strong spatial mixing for the hard-core model.

Strong spatial mixing holds for claw-free graphs.

Zero-free regions imply strong spatial mixing for graph homomorphism measures.

Abstract

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 implies that the associated probability measure, the \emph{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. We furthermore derive strong spatial mixing for graph homomorphism measures from absence of zeros of the graph homomorphism partition function.