For distributions over discrete product spaces $\prod_{i=1}^n \Omega_i'$, Glauber dynamics is a Markov chain that at each step, resamples a random coordinate conditioned on the other coordinates. We show that $k$-Glauber dynamics, which resamples a random subset of $k$ coordinates, mixes $k$ times faster in $\chi^2$-divergence, and assuming approximate tensorization of entropy, mixes $k$ times faster in KL-divergence. We apply this to obtain parallel algorithms in two settings: (1) For the Ising model $\mu_{J,h}(x)\propto \exp(\frac1 2\left\langle x,Jx \right\rangle + \langle h,x\rangle)$ with $\|J\|<1-c$ (the regime where fast mixing is known), we show that we can implement each step of $\widetilde \Theta(n/\|J\|_F)$-Glauber dynamics efficiently with a parallel algorithm, resulting in a parallel algorithm with running time $\widetilde O(\|J\|_F) = \widetilde O(\sqrt n)$. (2) For the mixed $p$-spin model at high enough temperature, we show that with high probability we show that we can implement each step of $\widetilde \Theta(\sqrt n)$-Glauber dynamics efficiently and obtain running time $\widetilde O(\sqrt n)$.

翻译：对于离散乘积空间$\prod_{i=1}^n \Omega_i'$上的分布，Glauber动力学是一种马尔可夫链，其在每一步基于其他坐标重新采样一个随机坐标。我们证明，$k$-Glauber动力学（即重新采样随机选取的$k$个坐标）在$\chi^2$散度下混合速度提升$k$倍，并且在熵近似可张量化的假设下，其KL散度下的混合速度亦提升$k$倍。我们将此应用于两个场景下的并行算法：（1）对于满足$\|J\|<1-c$（已知快速混合区域）的伊辛模型$\mu_{J,h}(x)\propto \exp(\frac1 2\left\langle x,Jx \right\rangle + \langle h,x\rangle)$，我们证明可通过并行算法高效实现$\widetilde \Theta(n/\|J\|_F)$-Glauber动力学的每一步，从而得到运行时间为$\widetilde O(\|J\|_F) = \widetilde O(\sqrt n)$的并行算法。（2）对于高温下的混合$p$自旋模型，我们以高概率证明可高效实现$\widetilde \Theta(\sqrt n)$-Glauber动力学的每一步，并获得运行时间$\widetilde O(\sqrt n)$。