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

2024-01-17

arXiv.org

AI Summary

This work establishes a universal connection between zero-free regions of the partition function for 2-spin systems and strong spatial mixing (SSM). To address multivariate partition functions with complex parameters $(eta,gamma,lambda)$, we introduce, for the first time, a Christoffel–Darboux-type identity on trees, combined with the Riemann mapping theorem to uniformly handle zero-free regions of arbitrary geometry—overcoming geometric constraints inherent in conventional tree-recursion methods. The framework accommodates non-uniform external fields and general pinning configurations. It systematically translates all known zero-free regions—including classical cases such as the Lee–Yang circle—into SSM guarantees. Moreover, we discover a novel notion of $pm$-spatial mixing and derive tight SSM bounds for non-uniform ferromagnetic Ising models. Our results achieve the first complete, unified, and geometry-agnostic translation from zero-location analysis to spatial mixing properties.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Social media analysis through the lenses of networks
• Web Mining and Content Analysis: Models for Web evolution

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 $(eta, gamma, lambda)$, and we allow the zero-free regions of $eta, gamma$ or $lambda$ 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 $eta, gamma$ or $lambda$ 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.

Problem

• Derive strong spatial mixing from zero-free regions
• Handle arbitrary shapes of zero-free regions
• Extend results to non-uniform external fields

Innovation

• Unified proof for SSM property
• Christoffel-Darboux identity application
• Riemann mapping for zero-free regions