From Zero-Freeness to Strong Spatial Mixing via a Christoffel-Darboux Type Identity

TL;DR

The paper develops a unifying method to deduce strong spatial mixing for the general 2-spin system from zero-free regions of the multivariate partition function $Z_G(\beta,\gamma,\lambda)$, extending to graphs with pinned vertices and non-uniform external fields. Central to the approach is a Christoffel-Darboux type identity on trees, which yields local dependence of coefficients and enables a transfer from zero-freeness to SSM via SAW-tree reductions and complex-analytic tools (Montel's theorem and the Riemann mapping theorem). The results cover zero-free regions near $\lambda=0$ and near $\beta\gamma=1$, and extend to non-uniform external fields, producing new SSM (and plus/minus spatial mixing) statements for the Ising model under Lee-Yang type conditions. The framework yields novel SSM implications beyond recurrence-based arguments and suggests broader applicability to non-uniform fields and, in extension, to $q$-spin systems. Overall, the work links analytic properties of the partition function to efficient approximation guarantees (SSM/Weitz-type FPTAS) for a wide class of spin systems on general graphs.

Abstract

We present a unifying proof to derive the strong spatial mixing (SSM) property for the general 2-spin system from zero-free regions of its partition function. Our proof works for the multivariate partition function over all three complex parameters $(β, γ, λ)$, and we allow the zero-free regions of $β, γ$ or $λ$ to be of arbitrary shapes. Our main technical contribution is to establish a Christoffel-Darboux type identity for the 2-spin system on trees so that we are able to handle zero-free regions of the three different parameters $β, γ$ or $λ$ in a unified way. We use Riemann mapping theorem to deal with zere-free regions of arbitrary shapes. Our result comprehensively turns all existing zero-free regions (to our best knowledge) of the partition function of the 2-spin system where pinned vertices are allowed into the SSM property. As a consequence, we obtain novel SSM properties for the 2-spin system beyond the direct argument for SSM based on tree recurrence. Moreover, we extend our result to handle the 2-spin system with non-uniform external fields. As an application, we obtain a new SSM property and two new forms of spatial mixing property, namely plus and minus spatial mixing for the non-uniform ferromagnetic Ising model from the celebrated Lee-Yang circle theorem.

Figures (4)

• Figure 1: Examples of regions for the 3 conditions in Theorem \ref{['thm:main']}
• Figure 2: Weitz's construction of SAW tree
• Figure :
• Figure :

Theorems & Definitions (71)

• Theorem 1
• Theorem 2
• Corollary 3
• Remark 1
• Theorem 4
• Corollary 5
• Theorem 6
• Theorem 7
• Definition 8: Feasible partial configuration
• Remark 2