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 $λ>0$ 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 $λ>0$ 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.

Figures (3)

• Figure 1: The components of the tree $T$.
• Figure 2: A commutative diagram displaying the relation between the $f_n$ and the $g_n$.
• Figure 3: How ratios propagate upwards in the tree towards a root vertex $v$ in $T_{H}$ (left) and in $P_{1}$ (right). Shown is $B_{\ell}(v)\subseteq H$. $\tau$ is a boundary condition on $S_H(v,\ell)$. In the left picture, $P_n$ examplifies a path that does not intersect the boundary $S_H(v,\ell)$, so $\tau$ has no influence along this path.

Theorems & Definitions (56)

• Definition 1: The tree of self avoiding walks
• Remark 2
• Lemma 3: Weitz Weitz2006CountingIndependentSetsuptotheTreeThreshold
• Definition 4
• Remark 5
• Remark 6
• Remark 7
• Theorem 8
• Definition 9
• Theorem 10