Let $μ$ be a probability distribution on a multi-state spin system on a set $V$ of sites; equivalently, a $d$-partite simplicial complex with distribution $μ$ on maximal faces. For any pair of vertices $u,v\in V$, define the pairwise spectral influence $\mathcal{I}_{u,v}$ as follows. Let $σ$ be a choice of spins $s_w\in S_w$ for every $w\in V\setminus\{u,v\}$, and construct a matrix in $\mathbb{R}^{(S_u\cup S_v)\times (S_u\cup S_v)}$ where for any $s_u\in S_u, s_v\in S_v$, 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 $u\to v$. As an application, we prove that the Glauber dynamics mixes rapidly up to (approximately) the phase transition for the multi-state hardcore model--a widely studied model in telecommunication networks and statistical physics (generalizing the hardcore model) introduced by Mazel and Suhov. As a by-product of our results, we also prove improved/almost optimal trickle-down theorems for partite simplicial complexes. Our proof builds on the trickle-down theorems via $\mathcal{C}$-Lorentzian polynomials machinery recently developed by the authors and Lindberg.

翻译：设$\mu$为集合$V$上多态自旋系统上的概率分布，或等价于以其最大面分布$\mu$定义的$d$部单纯复形。对于任意顶点对$u,v\in V$，定义成对谱影响$\mathcal{I}_{u,v}$如下：令$\sigma$为每个$w\in V\setminus\{u,v\}$的自旋选择$s_w\in S_w$，构造$\mathbb{R}^{(S_u\cup S_v)\times (S_u\cup S_v)}$中的矩阵，其中对于任意$s_u\in S_u, s_v\in S_v$，$(us_u,vs_v)$项为在给定$s_u$为$u$的自旋且给定$\sigma$的条件下$s_v$为$v$自旋的概率。则$\mathcal{I}_{u,v}$为所有$w\in V\setminus\{u,v\}$自旋选择下该矩阵的最大第二特征值。等价地，$\mathcal{I}_{u,v}$是包含每个$w\in V\setminus\{u,v\}$自旋的余维数为2的链接的最大局部谱展开。我们证明：若成对谱影响矩阵（元素为$\mathcal{I}_{u,v}$）的最大特征值远离1（即$\lambda_{\max}(\mathcal{I})\leq 1-ε$，且$X$连通），则格劳伯动力学快速混合并能从$\mu$中生成样本。该结果改进/推广了经典多布鲁申影响矩阵，因为$\mathcal{I}_{u,v}$是$u\to v$经典影响的下界。作为应用，我们证明对于多态硬核模型——Mazel与Suhov提出的电信网络与统计物理中广泛研究的模型（推广了硬核模型），格劳伯动力学在相变点附近（近似）快速混合。作为副产品，我们还证明了部单纯复形的改进/近似最优涓滴定理。证明基于作者和Lindberg近期发展的$\mathcal{C}$-洛伦兹多项式涓滴定理机制。