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On sampling two spin models using the local connective constant

Charilaos Efthymiou

TL;DR

Efthymiou develops optimum mixing bounds for Glauber dynamics on the Hard-core and Ising models by tying mixing to local graph structure through the radius-$k$ connective constant ${\upd}_{k}$ and the spectral data of the $k$-non-backtracking matrix ${\UpH}_{G,k}$. The approach blends Spectral Independence with deep tree-of-self-avoiding-walks constructions and extended influence matrices to obtain $O(n\log n)$ mixing times for fugacities below critical thresholds ${\lambda_c}({\upd}_{k})$ and stationary behavior controlled by ${\| (\UpH_{G,k})^N \|_2^{1/N}}$, with analogous results for the adjacency matrix ${\UpA}_G$. The work also provides hardness results indicating NP vs RP separations outside the favorable regimes and extends the framework to backtracking-walk analyses for broader graph classes, thereby unifying high-dimensional expander methods with classical Markov chain analyses. The results yield practically relevant mixing guarantees expressed in graph-local parameters, and advance the algorithmic understanding of sampling in Gibbs distributions on general graphs.

Abstract

This work establishes novel optimum mixing bounds for the Glauber dynamics on the Hard-core and Ising models. These bounds are expressed in terms of the local connective constant of the underlying graph $G$. This is a notion of effective degree for $G$. Our results have some interesting consequences for bounded degree graphs: (a) They include the max-degree bounds as a special case (b) They improve on the running time of the FPTAS considered in [Sinclair, Srivastava, \v Stefankoni\v c and Yin: PTRF 2017] for general graphs (c) They allow us to obtain mixing bounds in terms of the spectral radius of the adjacency matrix and improve on [Hayes: FOCS 2006]. We obtain our results using tools from the theory of high-dimensional expanders and, in particular, the Spectral Independence method [Anari, Liu, Oveis-Gharan: FOCS 2020]. We explore a new direction by utilising the notion of the $k$-non-backtracking matrix $H_{G,k}$ in our analysis with the Spectral Independence. The results with $H_{G,k}$ are interesting in their own right.

On sampling two spin models using the local connective constant

TL;DR

Efthymiou develops optimum mixing bounds for Glauber dynamics on the Hard-core and Ising models by tying mixing to local graph structure through the radius- connective constant and the spectral data of the -non-backtracking matrix . The approach blends Spectral Independence with deep tree-of-self-avoiding-walks constructions and extended influence matrices to obtain mixing times for fugacities below critical thresholds and stationary behavior controlled by , with analogous results for the adjacency matrix . The work also provides hardness results indicating NP vs RP separations outside the favorable regimes and extends the framework to backtracking-walk analyses for broader graph classes, thereby unifying high-dimensional expander methods with classical Markov chain analyses. The results yield practically relevant mixing guarantees expressed in graph-local parameters, and advance the algorithmic understanding of sampling in Gibbs distributions on general graphs.

Abstract

This work establishes novel optimum mixing bounds for the Glauber dynamics on the Hard-core and Ising models. These bounds are expressed in terms of the local connective constant of the underlying graph . This is a notion of effective degree for . Our results have some interesting consequences for bounded degree graphs: (a) They include the max-degree bounds as a special case (b) They improve on the running time of the FPTAS considered in [Sinclair, Srivastava, \v Stefankoni\v c and Yin: PTRF 2017] for general graphs (c) They allow us to obtain mixing bounds in terms of the spectral radius of the adjacency matrix and improve on [Hayes: FOCS 2006]. We obtain our results using tools from the theory of high-dimensional expanders and, in particular, the Spectral Independence method [Anari, Liu, Oveis-Gharan: FOCS 2020]. We explore a new direction by utilising the notion of the -non-backtracking matrix in our analysis with the Spectral Independence. The results with are interesting in their own right.

Paper Structure

This paper contains 66 sections, 43 theorems, 280 equations, 17 figures, 1 table.

Table of Contents

  1. Introduction
  2. Further results for the Ising model
  3. Results with backtracking walks
  4. Approach
  5. Spectral Bounds for ${\mathcal{I}}^{\Lambda,\tau}_{G}$ from likelihood ratios
  6. First attempt
  7. Second Attempt
  8. Structure of the paper.
  9. Preliminaries for the Analysis
  10. Glauber dynamics and mixing times.
  11. Spectral Independence
  12. (Local) Connective Constant:
  13. The adjacency matrix ${\UpA}_G$
  14. Matrices on oriented paths & PT-Invariance
  15. The $k$-non-backtracking matrix $\UpH_{G,k}$
  16. ...and 51 more sections

Key Result

Theorem 1.1

For any $\varepsilon \in (0,1)$, ${\Delta}>1$, $k\geq 1$ and $\upd_{k}>1$ consider graph $G=(V,E)$ of maximum degree ${\Delta}$ such that the radius-$k$ connective constant is $\upd_{k}$. Also, let $\mu$ be the Hard-core model on $G$ with fugacity $\lambda\leq (1-\varepsilon){\lambda_c}(\upd_{k})$.

Figures (17)

  • Figure 1: Initial graph $G$
  • Figure 2: $T_{\rm SAW}(G, w)$
  • Figure 3: Weighted Influences
  • Figure 4: $wb$-extension
  • Figure 5: $\{wb,ux\}$-extension
  • ...and 12 more figures

Theorems & Definitions (95)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Definition 3.1: $\delta$-contraction
  • Theorem 3.2: Adjacency Matrix
  • ...and 85 more