Contraction: a Unified Perspective of Correlation Decay and Zero-Freeness of 2-Spin Systems

TL;DR

This paper establishes a unified framework linking correlation decay and zero-freeness of the partition function in 2-spin systems, enabling efficient approximation algorithms for complex parameter regimes.

Contribution

It introduces the contraction property as a unified condition for correlation decay and zero-freeness, extending real parameters to complex neighborhoods for 2-spin systems.

Findings

Identifies new zero-free regions in complex parameter space.

Shows correlation decay persists in these regions.

Provides an FPTAS for partition functions on bounded degree graphs.

Abstract

We study complex zeros of the partition function of 2-spin systems, viewed as a multivariate polynomial in terms of the edge interaction parameters and the uniform external field. We obtain new zero-free regions in which all these parameters are complex-valued. Crucially based on the zero-freeness, we show the existence of correlation decay in these regions. As a consequence, we obtain an FPTAS for computing the partition function of 2-spin systems on graphs of bounded degree for these parameter settings. We introduce the contraction property as a unified sufficient condition to devise FPTAS via either Weitz's algorithm or Barvinok's algorithm. Our main technical contribution is a very simple but general approach to extend any real parameter of which the 2-spin system exhibits correlation decay to its complex neighborhood where the partition function is zero-free and correlation decay still exists. This result formally establishes the inherent connection between two distinct notions of phase transition for 2-spin systems: the existence of correlation decay and the zero-freeness of the partition function via a unified perspective, contraction.