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Counterexamples to a Weitz-Style Reduction for Multispin Systems

Kuikui Liu, Nitya Mani, Francisco Pernice

TL;DR

The paper investigates the feasibility of Weitz-style reductions for multispin ($q\ge3$) systems and proves fundamental obstacles: for a broad class of interactions, including ferromagnetic Potts, there is no universal external-field distribution that rounds belief-propagation images to product marginals, signaling that Weitz-type computation trees cannot straightforwardly extend to multispin models. It further provides positive evidence of convexity for the finite-temperature antiferromagnetic Potts model, suggesting the obstruction is not universal and depends on the ferromagnetic vs. antiferromagnetic nature. Together, these results imply that new ideas beyond Weitz’s gadget are required to achieve efficient algorithms for multispin systems, with potential implications for correlation decay, strong spatial mixing, and Glauber dynamics through alternative structural reductions or convexity phenomena.

Abstract

In a seminal paper, Weitz showed that for two-state spin systems, such as the Ising and hardcore models from statistical physics, correlation decay on trees implies correlation decay on arbitrary graphs. The key gadget in Weitz's reduction has been instrumental in recent advances in approximate counting and sampling, from analysis of local Markov chains like Glauber dynamics to the design of deterministic algorithms for estimating the partition function. A longstanding open problem in the field has been to find such a reduction for more general multispin systems like the uniform distribution over proper colorings of a graph. In this paper, we show that for a rich class of multispin systems, including the ferromagnetic Potts model, there are fundamental obstacles to extending Weitz's reduction to the multispin setting. A central component of our investigation is establishing nonconvexity of the image of the belief propagation functional, the standard tool for analyzing spin systems on trees. On the other hand, we provide evidence of convexity for the antiferromagnetic Potts model.

Counterexamples to a Weitz-Style Reduction for Multispin Systems

TL;DR

The paper investigates the feasibility of Weitz-style reductions for multispin () systems and proves fundamental obstacles: for a broad class of interactions, including ferromagnetic Potts, there is no universal external-field distribution that rounds belief-propagation images to product marginals, signaling that Weitz-type computation trees cannot straightforwardly extend to multispin models. It further provides positive evidence of convexity for the finite-temperature antiferromagnetic Potts model, suggesting the obstruction is not universal and depends on the ferromagnetic vs. antiferromagnetic nature. Together, these results imply that new ideas beyond Weitz’s gadget are required to achieve efficient algorithms for multispin systems, with potential implications for correlation decay, strong spatial mixing, and Glauber dynamics through alternative structural reductions or convexity phenomena.

Abstract

In a seminal paper, Weitz showed that for two-state spin systems, such as the Ising and hardcore models from statistical physics, correlation decay on trees implies correlation decay on arbitrary graphs. The key gadget in Weitz's reduction has been instrumental in recent advances in approximate counting and sampling, from analysis of local Markov chains like Glauber dynamics to the design of deterministic algorithms for estimating the partition function. A longstanding open problem in the field has been to find such a reduction for more general multispin systems like the uniform distribution over proper colorings of a graph. In this paper, we show that for a rich class of multispin systems, including the ferromagnetic Potts model, there are fundamental obstacles to extending Weitz's reduction to the multispin setting. A central component of our investigation is establishing nonconvexity of the image of the belief propagation functional, the standard tool for analyzing spin systems on trees. On the other hand, we provide evidence of convexity for the antiferromagnetic Potts model.

Paper Structure

This paper contains 15 sections, 14 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Background and Main Results
  3. Notation.
  4. On the Image of Belief Propagation
  5. Prior Works
  6. On Previous Computation Trees for Multispin Systems
  7. Proof of MainThm
  8. Evidence for Convexity of Fprod for Antiferromagnetic Potts
  9. Future Directions
  10. Belief propagation on general graphs
  11. Consequences of a Positive Answer to the Main Question
  12. Contraction Implies Correlation Decay for General Graphs
  13. Contraction Implies Spectral Independence and Optimal Mixing
  14. Signature analysis of B
  15. Verifying Numerical Inequalities

Key Result

Lemma 1.1

For every $q \in {\mathbb{N}}$, symmetric interaction matrix $A \in {\mathbb{R}}_{\geq0}^{q \times q}$, graph $G=(V,E)$, and vertex $v \in V$ with $d$ neighbors $N(v) = \{u_{1},\dots,u_{d}\}$, we have where the function $F_{A,d}$ is defined as follows: for an arbitrary distribution $\nu$ over $[q]^{d}$, we have

Theorems & Definitions (47)

  • Conjecture 1: Informal; Jerrum Jer95
  • Definition 1
  • Lemma 1.1: Recursion for General Graphs
  • Remark 1
  • Proposition 1.2: Wei06
  • Theorem 1.3
  • Definition 2
  • Theorem 1.4
  • Remark 2
  • Definition 1
  • ...and 37 more