Core Concepts
Zero-freeness of the partition function of the general 2-spin system implies strong spatial mixing on the corresponding parameter regions.
Abstract
The paper presents a unifying approach to derive the strong spatial mixing (SSM) property for the general 2-spin system from zero-free regions of its partition function. The approach works for the multivariate partition function over all three complex parameters (β, γ, λ), and allows the zero-free regions of β, γ or λ to be of arbitrary shapes, as long as they contain certain specific points.
The key technical contribution is the establishment of a Christoffel-Darboux type identity for the 2-spin system on trees. This identity plays an important role in the approach and enables the authors to prove that zero-freeness implies two key properties: local dependence of coefficients (LDC) and uniform bound on a circle. These two properties are then used to show that zero-freeness implies SSM.
The approach comprehensively turns all existing zero-free regions of the 2-spin system partition function (where pinned vertices are allowed) into the SSM property. As a consequence, new SSM results are obtained beyond the direct argument for SSM based on tree recurrence. The approach is also extended to handle the 2-spin system with non-uniform external fields, leading to a new SSM result for the non-uniform ferromagnetic Ising model from the Lee-Yang circle theorem.