Contraction: a Unified Perspective of Correlation Decay and Zero-Freeness of 2-Spin Systems
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2019
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| author | Shao, Shuai Sun, Yuxin |
| author_facet | Shao, Shuai Sun, Yuxin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1909_04244 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Contraction: a Unified Perspective of Correlation Decay and Zero-Freeness of 2-Spin Systems Shao, Shuai Sun, Yuxin Mathematical Physics Statistical Mechanics Data Structures and Algorithms 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. |
| title | Contraction: a Unified Perspective of Correlation Decay and Zero-Freeness of 2-Spin Systems |
| topic | Mathematical Physics Statistical Mechanics Data Structures and Algorithms |
| url | https://arxiv.org/abs/1909.04244 |