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Rapid Mixing via Coupling Independence for Spin Systems with Unbounded Degree

Xiaoyu Chen, Weiming Feng

Abstract

We develop a new framework to prove the mixing or relaxation time for the Glauber dynamics on spin systems with unbounded degree. It works for general spin systems including both $2$-spin and multi-spin systems. As applications for this approach: $\bullet$ We prove the optimal $O(n)$ relaxation time for the Glauber dynamics of random $q$-list-coloring on an $n$-vertices triangle-tree graph with maximum degree $Δ$ such that $q/Δ> α^\star$, where $α^\star \approx 1.763$ is the unique positive solution of the equation $α= \exp(1/α)$. This improves the $n^{1+o(1)}$ relaxation time for Glauber dynamics obtained by the previous work of Jain, Pham, and Vuong (2022). Besides, our framework can also give a near-linear time sampling algorithm under the same condition. $\bullet$ We prove the optimal $O(n)$ relaxation time and near-optimal $\widetilde{O}(n)$ mixing time for the Glauber dynamics on hardcore models with parameter $λ$ in $\textit{balanced}$ bipartite graphs such that $λ< λ_c(Δ_L)$ for the max degree $Δ_L$ in left part and the max degree $Δ_R$ of right part satisfies $Δ_R = O(Δ_L)$. This improves the previous result by Chen, Liu, and Yin (2023). At the heart of our proof is the notion of $\textit{coupling independence}$ which allows us to consider multiple vertices as a huge single vertex with exponentially large domain and do a "coarse-grained" local-to-global argument on spin systems. The technique works for general (multi) spin systems and helps us obtain some new comparison results for Glauber dynamics.

Rapid Mixing via Coupling Independence for Spin Systems with Unbounded Degree

Abstract

We develop a new framework to prove the mixing or relaxation time for the Glauber dynamics on spin systems with unbounded degree. It works for general spin systems including both -spin and multi-spin systems. As applications for this approach: We prove the optimal relaxation time for the Glauber dynamics of random -list-coloring on an -vertices triangle-tree graph with maximum degree such that , where is the unique positive solution of the equation . This improves the relaxation time for Glauber dynamics obtained by the previous work of Jain, Pham, and Vuong (2022). Besides, our framework can also give a near-linear time sampling algorithm under the same condition. We prove the optimal relaxation time and near-optimal mixing time for the Glauber dynamics on hardcore models with parameter in bipartite graphs such that for the max degree in left part and the max degree of right part satisfies . This improves the previous result by Chen, Liu, and Yin (2023). At the heart of our proof is the notion of which allows us to consider multiple vertices as a huge single vertex with exponentially large domain and do a "coarse-grained" local-to-global argument on spin systems. The technique works for general (multi) spin systems and helps us obtain some new comparison results for Glauber dynamics.
Paper Structure (33 sections, 36 theorems, 107 equations, 1 algorithm)

This paper contains 33 sections, 36 theorems, 107 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Applications
  4. Technical Results and Proof Overview
  5. Coupling Independence
  6. Compare Markov Chains via Coupling Independence
  7. Proof Sketch of \ref{['main:coloring']}
  8. Proof of \ref{['thm:Trel-from-conj']}
  9. Establish Coupling Independence
  10. Proof Overview
  11. Preliminaries
  12. -Divergences and -Entropies
  13. Markov Chain Background
  14. Block dynamics
  15. Coupling and Path Coupling
  16. ...and 18 more sections

Key Result

Theorem 1

Let $\delta > 0$ be a constant. For any triangle-free graph $G=(V,E)$ and color lists $(L_v)_{v \in V}$, if $|L_v| \geq (\alpha^\star + \delta)\Delta$ for all $v \in V$, where $\Delta \geq 3$ is the maximum degree of $G$, then relaxation time of Glauber dynamics is $O_\delta(n)$, where $n$ is the nu

Theorems & Definitions (64)

  • Theorem 1: Coloring: Relaxation Time
  • Conjecture 2: Folklore
  • Proposition 3
  • Theorem 4: Coloring: Algorithm
  • Theorem 5: Bipartite Hardcore: Relaxation Time
  • Theorem 6: Bipartite Hardcore: Mixing Time
  • Definition 7: Coupling Independence
  • Definition 8: Relaxation Time with Pinning
  • Theorem 9: Relaxation Time Comparison
  • Remark 10: Hardcore Model in Uniqueness Regime
  • ...and 54 more