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

Abstract

Let $\hat{I}¼$ be a probability distribution on a multi-state spin system on a set $V$ of sites; equivalently, a $d$-partite simplicial complex with distribution $\hat{I}¼$ on maximal faces. For any pair of vertices $u,v\in V$, define the pairwise spectral influence ${\mathcal{I}}_{u,v}$ as follows. Let $\ddot{I}ƒ$ 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 $(u{s}_u,v{s}_v)$-entry is the probability that $s_v$ is the spin of $v$ conditioned on $s_u$ being the spin of $u$ and on $\ddot{I}ƒ$. 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. $\hat{I}{»}_{\max }(\mathcal{I})\le 1-\hat{I}\mu$ (and $X$ is connected), then the Glauber dynamics mixes rapidly and generate samples from $\hat{I}¼$. 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.