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Spectral Independence Beyond Total Influence on Trees and Related Graphs

Xiaoyu Chen, Xiongxin Yang, Yitong Yin, Xinyuan Zhang

TL;DR

The paper introduces a direct, approximate-inverse approach to spectral independence that bypasses total-influence bounds, enabling constant upper bounds on λ_max(Ψ_μ) for Gibbs distributions where total influence is unbounded. It applies the method to the monomer-dimer model on trees and graphs with large girth, deriving explicit SI bounds that translate into near-optimal Glauber-gap results, independent of maximum degree Δ. For the hardcore model on trees, the approach yields spectral-independence up to λ < e^2 with a quantified bound and corresponding Ω(n^{-1}) mixing, and it illuminates the boundary where SI becomes unbounded as λ grows. Together, these results advance understanding of fast sampling via Glauber dynamics in high-degree or locally-tree-like graphs, and they provide a framework to lift tree-based SI bounds to larger graphs under girth conditions, with explicit trade-offs between girth, Δ, and model parameters.

Abstract

We study how to establish $\textit{spectral independence}$, a key concept in sampling, without relying on total influence bounds, by applying an $\textit{approximate inverse}$ of the influence matrix. Our method gives constant upper bounds on spectral independence for two foundational Gibbs distributions known to have unbounded total influences: $\bullet$ The monomer-dimer model on graphs with large girth (including trees). Prior to our work, such results were only known for graphs with constant maximum degrees or infinite regular trees, as shown by Chen, Liu, and Vigoda (STOC '21). $\bullet$ The hardcore model on trees with fugacity $λ< \mathrm{e}^2$. This remarkably surpasses the well-known $λ_r>\mathrm{e}-1$ lower bound for the reconstruction threshold on trees, significantly improving upon the current threshold $λ< 1.3$, established in a prior work by Efthymiou, Hayes, Štefankovič, and Vigoda (RANDOM '23). Consequently, we establish optimal $Ω(n^{-1})$ spectral gaps of the Glauber dynamics for these models on arbitrary trees, regardless of the maximum degree $Δ$.

Spectral Independence Beyond Total Influence on Trees and Related Graphs

TL;DR

The paper introduces a direct, approximate-inverse approach to spectral independence that bypasses total-influence bounds, enabling constant upper bounds on λ_max(Ψ_μ) for Gibbs distributions where total influence is unbounded. It applies the method to the monomer-dimer model on trees and graphs with large girth, deriving explicit SI bounds that translate into near-optimal Glauber-gap results, independent of maximum degree Δ. For the hardcore model on trees, the approach yields spectral-independence up to λ < e^2 with a quantified bound and corresponding Ω(n^{-1}) mixing, and it illuminates the boundary where SI becomes unbounded as λ grows. Together, these results advance understanding of fast sampling via Glauber dynamics in high-degree or locally-tree-like graphs, and they provide a framework to lift tree-based SI bounds to larger graphs under girth conditions, with explicit trade-offs between girth, Δ, and model parameters.

Abstract

We study how to establish , a key concept in sampling, without relying on total influence bounds, by applying an of the influence matrix. Our method gives constant upper bounds on spectral independence for two foundational Gibbs distributions known to have unbounded total influences: The monomer-dimer model on graphs with large girth (including trees). Prior to our work, such results were only known for graphs with constant maximum degrees or infinite regular trees, as shown by Chen, Liu, and Vigoda (STOC '21). The hardcore model on trees with fugacity . This remarkably surpasses the well-known lower bound for the reconstruction threshold on trees, significantly improving upon the current threshold , established in a prior work by Efthymiou, Hayes, Štefankovič, and Vigoda (RANDOM '23). Consequently, we establish optimal spectral gaps of the Glauber dynamics for these models on arbitrary trees, regardless of the maximum degree .
Paper Structure (26 sections, 40 theorems, 117 equations)

This paper contains 26 sections, 40 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. The monomer-dimer model.
  3. The hardcore model.
  4. Related work
  5. High dimensional expander.
  6. (Matrix) trickle-down method.
  7. Lifting spectral independence from tree to graph.
  8. Glauber dynamics on tree.
  9. Preliminaries
  10. Notations and conventions
  11. Markov chains and mixing time
  12. Spectral independence and related matrices
  13. Spectral independence of $k$-transformed distributions
  14. Inverse of product distance matrix
  15. Linear algebra toolkit
  16. ...and 11 more sections

Key Result

Theorem 2

Let $T = (V, E)$ be a tree of $n$ vertices and $\lambda > 0$. The Gibbs distribution $\mu$ of the monomer-dimer model on $T$ with fugacity $\lambda$ has Moreover, the Glauber dynamics on $\mu$ has asymptotically optimal spectral gap $\Omega_\lambda(n^{-1})$.

Theorems & Definitions (91)

  • Conjecture 1: chen2021optimal
  • Theorem 2
  • Theorem 3
  • Example 4
  • Conjecture 5
  • Conjecture 6: efthymiou2023optimal
  • Theorem 7
  • Theorem 8
  • Definition 9: influence matrix
  • Definition 10: spectral independence anari2020spectral
  • ...and 81 more