Trickle-down Theorems via C-Lorentzian Polynomials II: Pairwise Spectral Influence and Improved Dobrushin's Condition

2025-10-07

AI Summary

This paper investigates rapid mixing of Glauber dynamics on multi-state spin systems, where the classical Dobrushin condition fails to adequately control dependencies in higher-order structures. Method: We introduce the notion of *pairwise spectral influence* (I_{u,v}), construct the spectral influence matrix, and characterize its largest eigenvalue (lambda_{max}) as the key determinant of mixing time. Our approach integrates C-Lorentzian polynomial theory, local spectral expansion techniques, and matrix analysis to establish a novel “trickle-down” theorem applicable to partial simplicial complexes. Contribution/Results: We prove that when (lambda_{max} < 1) and bounded away from 1, Glauber dynamics mixes rapidly at nearly optimal rate. This spectral condition is strictly weaker than standard Dobrushin-type assumptions, yielding a more general and tighter criterion for efficient sampling in high-dimensional probabilistic models.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Let $μ$ be a probability distribution on a multi-state spin system on a set $V$ of sites. Equivalently, we can think of this as a $d$-partite simplical complex with distribution $μ$ on maximal faces. For any pair of vertices $u,vin V$, define the pairwise spectral influence $mathcal{I}_{u,v}$ as follows. Let $σ$ be a choice of spins $s_win S_w$ for every $win V setminus {u,v}$, and construct a matrix in $mathbb{R}^{(S_ucup S_v) imes (S_ucup S_v)}$ where for any $s_uin S_u, s_vin S_v$, the $(us_u,vs_v)$-entry is the probability that $s_v$ is the spin of $v$ conditioned on $s_u$ being the spin of $u$ and on $σ$. Then $mathcal{I}_{u,v}$ is the maximal second eigenvalue of this matrix, over all choices of spins for all $w in V setminus {u,v}$. Equivalently, $mathcal{I}_{u,v}$ is the maximum local spectral expansion of links of codimension $2$ that include a spin for every $w in V setminus {u,v}$. We show that if the largest eigenvalue of the pairwise spectral influence matrix with entries $mathcal{I}_{u,v}$ is bounded away from 1, i.e. $λ_{max}(mathcal{I})leq 1-ε$ (and $X$ is connected), then the Glauber dynamics mixes rapidly and generate samples from $μ$. This improves/generalizes the classical Dobrushin's influence matrix as the $mathcal{I}_{u,v}$ lower-bounds the classical influence of $u o v$. As a by-product, we also prove improved/almost optimal trickle-down theorems for partite simplicial complexes. The proof builds on the trickle-down theorems via $mathcal{C}$-Lorentzian polynomials machinery recently developed by the authors and Lindberg.

Problem

• Analyzing pairwise spectral influence in multi-state spin systems
• Improving Dobrushin's condition for rapid mixing of Glauber dynamics
• Proving enhanced trickle-down theorems for partite simplicial complexes

Innovation

• Defining pairwise spectral influence for spin systems
• Bounding spectral influence matrix eigenvalues for mixing
• Using C-Lorentzian polynomials for trickle-down theorems