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Planting and MCMC Sampling from the Potts model

Andreas Galanis, Leslie Ann Goldberg, Paulina Smolarova

Abstract

We consider the problem of sampling from the ferromagnetic $q$-state Potts model on the random $d$-regular graph with parameter $β>0$. A key difficulty that arises in sampling from the model is the existence of a metastability window $(β_u,β_u')$ where the distribution has two competing modes, the so-called disordered and ordered phases, causing MCMC-based algorithms to be slow mixing from worst-case initialisations. To this end, Helmuth, Jenssen and Perkins designed a sampling algorithm that works for all $β$ when $q$ is large, using cluster expansion methods; more recently, their analysis technique has been adapted to show that random-cluster dynamics mixes fast when initialised more judiciously. However, a bottleneck behind cluster-expansion arguments is that they inherently only work for large $q$, whereas it is widely conjectured that sampling is possible for all $q,d\geq 3$. The only result so far that applies to general $q,d\geq 3$ is by Blanca and Gheissari who showed that the random-cluster dynamics mixes fast for $β<β_u$. For $β>β_u$, certain correlation phenomena emerge because of the metastability which have been hard to handle, especially for small $q$ and $d$. Our main contribution is to perform a delicate analysis of the Potts distribution and the random-cluster dynamics that goes beyond the threshold $β_u$. We use planting as the main tool in our proofs, and combine it with the analysis of random-cluster dynamics. We are thus able to show that the random-cluster dynamics initialised from all-out mixes fast for all integers $q,d\geq 3$ beyond the uniqueness threshold $β_u$; our analysis works all the way up to the threshold $β_c\in (β_u,β_u')$ where the dominant mode switches from disordered to ordered. We also obtain an algorithm in the ordered regime $β>β_c$ that refines significantly the range of $q,d$.

Planting and MCMC Sampling from the Potts model

Abstract

We consider the problem of sampling from the ferromagnetic -state Potts model on the random -regular graph with parameter . A key difficulty that arises in sampling from the model is the existence of a metastability window where the distribution has two competing modes, the so-called disordered and ordered phases, causing MCMC-based algorithms to be slow mixing from worst-case initialisations. To this end, Helmuth, Jenssen and Perkins designed a sampling algorithm that works for all when is large, using cluster expansion methods; more recently, their analysis technique has been adapted to show that random-cluster dynamics mixes fast when initialised more judiciously. However, a bottleneck behind cluster-expansion arguments is that they inherently only work for large , whereas it is widely conjectured that sampling is possible for all . The only result so far that applies to general is by Blanca and Gheissari who showed that the random-cluster dynamics mixes fast for . For , certain correlation phenomena emerge because of the metastability which have been hard to handle, especially for small and . Our main contribution is to perform a delicate analysis of the Potts distribution and the random-cluster dynamics that goes beyond the threshold . We use planting as the main tool in our proofs, and combine it with the analysis of random-cluster dynamics. We are thus able to show that the random-cluster dynamics initialised from all-out mixes fast for all integers beyond the uniqueness threshold ; our analysis works all the way up to the threshold where the dominant mode switches from disordered to ordered. We also obtain an algorithm in the ordered regime that refines significantly the range of .

Paper Structure

This paper contains 24 sections, 27 theorems, 71 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Results
  3. Proof outline
  4. Further Discussion
  5. Preliminaries
  6. The configuration model
  7. The random-cluster dynamics
  8. Phases of the Potts model
  9. Planting disordered and ordered configurations
  10. Mixing within the disordered and ordered phases for the RC model
  11. Phases for the random cluster model via Potts
  12. Mixing and Weak Spatial Mixing (WSM) within a phase
  13. Proof for WSM in the disordered regime
  14. Proof for WSM within the ordered regime
  15. WSM within the ordered phase
  16. ...and 9 more sections

Key Result

Theorem 1.1

For integers $d,q\geq 3$ and real $\beta\in (0,\beta_c)$, the following holds whp over $\mathbb{G}=\mathbb{G}_{n,d}$. There is an MCMC algorithm that, on input $\mathbb{G}$ and $\varepsilon\geq \mathrm{e}^{-\Omega(n)}$, outputs in $O(n\log n\log(\frac{1}{\varepsilon}))$ steps a sample $\hat{\sigma}\

Figures (1)

  • Figure 1: The exploration process of the percolated planted graph used in the proof of Lemma \ref{['lem:dispath']}. This is a combination of exposing first an edge in the planted configuration model, captured by the routine Match$(\cdot,\cdot)$, followed by a percolation step with probability $p=1-\mathrm{e}^{-\beta}$ whenever the edge is monochromatic.

Theorems & Definitions (52)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1: BIShard
  • Lemma 2.2
  • Lemma 2.3: coja2023
  • Definition 3.1: The ordered/disordered phases
  • Corollary 3.2
  • proof : Proof of Corollary \ref{['cor:smallgraphb']}
  • Lemma 3.3
  • Theorem 3.4
  • ...and 42 more