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

arXiv: https://arxiv.org/abs/1909.04244 · HTML: https://mlsanity.com/paper/1909.04244/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Sep 2019.

## Built on by (most cited first, 1 of 1)

- [Rapid mixing in positively weighted restricted Boltzmann machines](https://mlsanity.com/paper/2604.00963/) (2026)

## Abstract (arXiv)

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.