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Cutoff for the Swendsen-Wang dynamics on the complete graph

Antonio Blanca, Zhezheng Song

TL;DR

This work resolves the mixing-time behavior of Swendsen–Wang dynamics for the $q$-state mean-field Potts model on the complete graph in the low-temperature regime $β>q$. By constructing a multi-phase coupling and introducing a $q\times q$ projection to the majority vector, the authors prove a sharp cutoff: the mixing time is $τ_{\rm mix}^{\mathrm{SW}} = c(β,q) \log n + Θ(1)$, with the constant $c(β,q)$ depending explicitly on the derivative at the drift-function fixed point $a(β,q)$. The analysis hinges on the drift function $F$ governing the evolution of the largest color class, percolation-based random-graph controls for the SW steps, and a projection-chain coupling that aligns color counts across copies. These techniques yield a precise, finite-window cutoff result in a regime where previous work only showed $Θ(\log n)$ growth, highlighting both the dynamics' sharp transition and its algorithmic implications for sampling from the Potts model on complete graphs.

Abstract

We study the speed of convergence of the Swendsen-Wang (SW) dynamics for the $q$-state ferromagnetic Potts model on the $n$-vertex complete graph, known as the mean-field model. The SW dynamics was introduced as an attractive alternative to the local Glauber dynamics, often offering faster convergence rates to stationarity in a variety of settings. A series of works have characterized the asymptotic behavior of the speed of convergence of the mean-field SW dynamics for all $q \ge 2$ and all values of the inverse temperature parameter $β> 0$. In particular, it is known that when $β> q$ the mixing time of the SW dynamics is $Θ(\log n)$. We strengthen this result by showing that for all $β> q$, there exists a constant $c(β,q) > 0$ such that the mixing time of the SW dynamics is $c(β,q) \log n + Θ(1)$. This implies that the mean-field SW dynamics exhibits the cutoff phenomenon in this temperature regime, demonstrating that this Markov chain undergoes a sharp transition from ''far from stationarity'' to ''well-mixed'' within a narrow $Θ(1)$ time window. The presence of cutoff is algorithmically significant, as simulating the chain for fewer steps than its mixing time could lead to highly biased samples.

Cutoff for the Swendsen-Wang dynamics on the complete graph

TL;DR

This work resolves the mixing-time behavior of Swendsen–Wang dynamics for the -state mean-field Potts model on the complete graph in the low-temperature regime . By constructing a multi-phase coupling and introducing a projection to the majority vector, the authors prove a sharp cutoff: the mixing time is , with the constant depending explicitly on the derivative at the drift-function fixed point . The analysis hinges on the drift function governing the evolution of the largest color class, percolation-based random-graph controls for the SW steps, and a projection-chain coupling that aligns color counts across copies. These techniques yield a precise, finite-window cutoff result in a regime where previous work only showed growth, highlighting both the dynamics' sharp transition and its algorithmic implications for sampling from the Potts model on complete graphs.

Abstract

We study the speed of convergence of the Swendsen-Wang (SW) dynamics for the -state ferromagnetic Potts model on the -vertex complete graph, known as the mean-field model. The SW dynamics was introduced as an attractive alternative to the local Glauber dynamics, often offering faster convergence rates to stationarity in a variety of settings. A series of works have characterized the asymptotic behavior of the speed of convergence of the mean-field SW dynamics for all and all values of the inverse temperature parameter . In particular, it is known that when the mixing time of the SW dynamics is . We strengthen this result by showing that for all , there exists a constant such that the mixing time of the SW dynamics is . This implies that the mean-field SW dynamics exhibits the cutoff phenomenon in this temperature regime, demonstrating that this Markov chain undergoes a sharp transition from ''far from stationarity'' to ''well-mixed'' within a narrow time window. The presence of cutoff is algorithmically significant, as simulating the chain for fewer steps than its mixing time could lead to highly biased samples.

Paper Structure

This paper contains 13 sections, 16 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Couplings
  4. Random graph lemmas
  5. Drift function
  6. Proof of main result: cutoff when $\beta > q$
  7. Convergence to the neighborhood of the majority vector
  8. Proofs of Lemmas \ref{['lemma:main:contract']}, \ref{['lemma:f:fp-convergence']}, and \ref{['lemma:step:concentration']}
  9. Mixing time lower bound
  10. Spin fraction coalescence in partition
  11. Projection chain
  12. Proof of facts about the drift function $F$
  13. Proof of Lemma \ref{['lemma:potts:conc']}

Key Result

Theorem 1.1

Let $q \ge 2$ and $\beta > \beta_{\textsc{r}}=q$. Then, there exists a constant $c(\beta,q) > 0$ such that SW dynamics for the $q$-state mean-field ferromagnetic Potts model exhibits cutoff at mixing time $\tau_{\rm mix}^{\mathrm{SW}} = c(\beta,q) \log n$ with cutoff window $\Theta(1)$; that is, $\t

Theorems & Definitions (32)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1: Lemma 5.7 LNNP
  • Lemma 2.2: Lemma 5.4 LNNP
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • ...and 22 more