Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Shuai Shao, Yuxin Sun

TL;DR

This work treats $Z_G(\beta,\gamma,\lambda)$ as a multivariate complex polynomial and identifies new zero-free regions where all three parameters are complex. It proves correlation decay (strong spatial mixing) persists in these regions and gives an FPTAS for bounded-degree graphs via Weitz or Barvinok methods. A central concept, contraction, unifies zero-freeness and correlation decay, and the authors extend real-contraction results to complex neighborhoods using analytic continuation and the inverse function theorem. They explicitly define four real-parameter sets $\mathcal{S}^{\Delta}_i$ that admit real contraction and show a neighborhood around each point preserves zero-freeness and decay. The work thus links two notions of phase transition in 2-spin systems in a unified, algorithmically actionable way.

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.

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

TL;DR

This work treats as a multivariate complex polynomial and identifies new zero-free regions where all three parameters are complex. It proves correlation decay (strong spatial mixing) persists in these regions and gives an FPTAS for bounded-degree graphs via Weitz or Barvinok methods. A central concept, contraction, unifies zero-freeness and correlation decay, and the authors extend real-contraction results to complex neighborhoods using analytic continuation and the inverse function theorem. They explicitly define four real-parameter sets that admit real contraction and show a neighborhood around each point preserves zero-freeness and decay. The work thus links two notions of phase transition in 2-spin systems in a unified, algorithmically actionable way.

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.

Paper Structure

This paper contains 12 sections, 16 theorems, 45 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Weitz's Algorithm
  3. Notations and definitions
  4. Correlation decay and real contraction
  5. Barvinok's Algorithm
  6. Zero-freeness and complex contraction
  7. From Real Contraction to Complex Contraction
  8. Appendix
  9. Self-Avoiding Walk Tree
  10. Proof of Lemma \ref{['lem:c1c2']}
  11. Proof of Lemma \ref{['complex-to-zerofree']}
  12. Proof of Lemma \ref{['lem:complex-to-correlation']}

Key Result

Theorem 1.1

Fix $\Delta \in \mathbb{N}$. If $\pmb\zeta_0\in \mathbb{R}^3$ satisfies real contraction for $\Delta$, then there exists a $\delta>0$ such that for any $\pmb \zeta \in \mathbb{C}^3$ where $\|\pmb\zeta-\pmb\zeta_0\|_{\infty}<\delta$, we have As a consequence, there is an FPTAS for computing $Z_{G}(\pmb\zeta)$.

Figures (3)

  • Figure 1: The structure of our approach
  • Figure 2: Commutative diagram between $F$ and $F^{\varphi}$
  • Figure 3: Weitz's SAW tree construction

Theorems & Definitions (30)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 2.1: Recursion function
  • Definition 2.2: Feasible configuration
  • Definition 2.3: Strong spatial mixing
  • Definition 2.4: Real contraction
  • Lemma 2.5
  • proof
  • Definition 2.6: Uniqueness condition lly13
  • ...and 20 more