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

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.