Let $\mu$ 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 $\mu$ on maximal faces. For any pair of vertices $u,v\in V$, define the pairwise spectral influence $\mathcal{I}_{u,v}$ as follows. Let $\sigma$ 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 $\sigma$. 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. $\lambda_{\max}(\mathcal{I})\leq 1-\epsilon$ (and $X$ is connected), then the Glauber dynamics mixes rapidly and generate samples from $\mu$. 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 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.

翻译：设 $\mu$ 为定义在顶点集 $V$ 上一个多态自旋系统的概率分布。等价地，可将其视为一个具有最大面分布 $\mu$ 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 $\sigma$ 为所有 $w\in V \setminus \{u,v\}$ 的自旋选择 $s_w\in S_w$，构造矩阵 $\mathbb{R}^{(S_u\cup S_v)\times (S_u\cup S_v)}$，其中对任意 $s_u\in S_u, s_v\in S_v$，元素 $(us_u,vs_v)$ 表示在给定 $u$ 的自旋为 $s_u$ 且给定 $\sigma$ 的条件下 $v$ 的自旋为 $s_v$ 的条件概率。则 $\mathcal{I}_{u,v}$ 定义为该矩阵在所有 $w \in V \setminus \{u,v\}$ 自旋选择下的最大第二特征值。等价地，$\mathcal{I}_{u,v}$ 是包含所有 $w \in V \setminus \{u,v\}$ 自旋的余维2链环的最大局部谱展开。本文证明：若以 $\mathcal{I}_{u,v}$ 为元素的成对谱影响矩阵的最大特征值有界远离1（即 $\lambda_{\max}(\mathcal{I})\leq 1-\epsilon$）且 $X$ 连通，则Glauber动力学会快速混合并生成服从 $\mu$ 的样本。该结论改进并推广了经典的Dobrushin影响矩阵，因为 $\mathcal{I}_{u,v}$ 构成了经典影响 $u\to v$ 的下界。作为推论，我们还证明了关于部单纯复形的改进/近乎最优的滴流定理。证明基于作者与Lindberg近期通过 $\mathcal{C}$-Lorentzian多项式体系建立的滴流定理。