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

Authors Shuai Shao , Yuxin Sun

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Shuai Shao
  • Department of Computer Sciences, University of Wisconsin-Madison, Madison, WI, USA
Yuxin Sun
  • Department of Computer Sciences, University of Wisconsin-Madison, Madison, WI, USA


We want to thank Professor Jin-Yi Cai, the advisor of the first author, for many inspiring discussions and valuable comments on a preliminary version of this paper. Despite his support, he generously declined our invitation for co-authorship. We also want to thank Professor Alex Scott, Dr. Jingcheng Liu and anonymous reviewers for their helpful comments.

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Shuai Shao and Yuxin Sun. Contraction: A Unified Perspective of Correlation Decay and Zero-Freeness of 2-Spin Systems. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 96:1-96:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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 are able to extend the existence of correlation decay to these complex regions from real parameters. 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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Approximation algorithms
  • 2-Spin system
  • Correlation decay
  • Zero-freeness
  • Phase transition
  • Contraction


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