We study the mixing time of Glauber dynamics on monotone systems. For monotone systems satisfying the entropic independence condition, we prove a new mixing time comparison result for Glauber dynamics. For concrete applications, we obtain \(\tilde O(n)\) mixing time for the random cluster model induced by the ferromagnetic Ising model with consistently biased external fields, and \(\tilde O(n^2)\) mixing time for the bipartite hardcore model under the one-sided uniqueness condition, where \(n\) is the number of variables in corresponding models, improving the best known results in [Chen and Zhang, SODA’23] and [Chen, Liu, and Yin, FOCS’23], respectively.

Spin systems form an important class of undirected graphical models. For two Gibbs distributions $\mu$ and $\nu$ induced by two spin systems on the same graph $G = (V, E)$, we study the problem of approximating the total variation distance $d_{\mathrm{TV}}\left({\mu},{\nu}\right)$ with an $\epsilon$-relative error. We propose a new reduction that connects the problem of approximating the TV-distance to sampling and approximate counting. Our applications include the hardcore model and the antiferromagnetic Ising model in the uniqueness regime, the ferromagnetic Ising model, and the general Ising model satisfying the spectral condition. Additionally, we explore the computational complexity of approximating the total variation distance $d_{\mathrm{TV}}\left({\mu_S},{\nu_S}\right)$ between two marginal distributions on an arbitrary subset $S \subseteq V$. We prove that this problem remains hard even when both $\mu$ and $\nu$ admit polynomial-time sampling and approximate counting algorithms.