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Near optimal bounds for weak and strong spatial mixing for the anti-ferromagnetic Potts model on trees

Ferenc Bencs, Khallil Berrekkal, Guus Regts

TL;DR

This work analyzes the anti-ferromagnetic Potts model on rooted trees, establishing near-optimal weak and strong spatial mixing (WSM/SSM) bounds in terms of the weight parameter $w$, the number of colors $q$, and the branching factor $d$. The authors develop a square-root ratio coordinate system and a cavity-like recursion that express root marginals through its children, and prove contraction of this recursion under a near-threshold range of $w$ using inductive arguments. They derive explicit bounds on local weights $\lambda_v$ and marginals, enabling precise contraction factors and resulting in concrete WSM/SSM regimes that closely approach conjectured thresholds as $d$ grows. These results extend previous $w=0$ and large-$d$ findings to a broader, near-optimal parameter range, with implications for efficient sampling and Glauber dynamics on graphs with large girth or bounded degree.

Abstract

We show that the anti-ferromagnetic Potts model on trees exhibits strong spatial mixing for a near-optimal range of parameters. Our work complements recent results of Chen, Liu, Mani, and Moitra [arXiv.2304.01954] who showed this to be true in the infinite temperature setting, corresponding to uniform proper colorings. We furthermore prove weak spatial mixing results complementing results in [arXiv.2304.01954].

Near optimal bounds for weak and strong spatial mixing for the anti-ferromagnetic Potts model on trees

TL;DR

This work analyzes the anti-ferromagnetic Potts model on rooted trees, establishing near-optimal weak and strong spatial mixing (WSM/SSM) bounds in terms of the weight parameter , the number of colors , and the branching factor . The authors develop a square-root ratio coordinate system and a cavity-like recursion that express root marginals through its children, and prove contraction of this recursion under a near-threshold range of using inductive arguments. They derive explicit bounds on local weights and marginals, enabling precise contraction factors and resulting in concrete WSM/SSM regimes that closely approach conjectured thresholds as grows. These results extend previous and large- findings to a broader, near-optimal parameter range, with implications for efficient sampling and Glauber dynamics on graphs with large girth or bounded degree.

Abstract

We show that the anti-ferromagnetic Potts model on trees exhibits strong spatial mixing for a near-optimal range of parameters. Our work complements recent results of Chen, Liu, Mani, and Moitra [arXiv.2304.01954] who showed this to be true in the infinite temperature setting, corresponding to uniform proper colorings. We furthermore prove weak spatial mixing results complementing results in [arXiv.2304.01954].
Paper Structure (13 sections, 20 theorems, 118 equations)

This paper contains 13 sections, 20 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Motivation from computer science
  3. Approach and organization
  4. Setup and detailed approach
  5. A change of coordinates and a recursion
  6. Inductive proof of WSM
  7. Inductive proof of SSM
  8. A bound on the strength of the contraction
  9. Bounds on the local weights
  10. Bounds on the marginal of the root vertex
  11. Further useful bounds
  12. Bounding the local weights
  13. Concluding remarks and questions

Key Result

Lemma 1

Let $q>0$ be an integer. The $q$-state Potts model at parameter $w\geq 0$ exhibits weak spatial mixing with exponential decay rate $r\in (0,1)$ on a family of rooted trees $\mathcal{T}$, if there exists a constant $C>0$ such that for each rooted tree $(T,v)\in \mathcal{T}$, and any two boundary cond

Theorems & Definitions (41)

  • Definition 1: Weak Spatial Mixing (WSM)
  • Definition 2: Strong Spatial Mixing (SSM)
  • Lemma 1
  • proof
  • Theorem 2
  • Remark 1
  • Remark 2
  • Theorem 3
  • Lemma 4
  • proof
  • ...and 31 more