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 Ising models $\mu_{J,h}(x)\propto \exp(\frac1 2\left\langle x,Jx \right\rangle + \langle h,x\rangle)$ with $\|J\|<1-c$ (the regime where fast mixing is known), where we show that we can implement each step of $\widetilde O(n/\|J\|_F)$-Glauber dynamics efficiently with a parallel algorithm, resulting in a parallel algorithm with running time $\widetilde O(\|J\|_F) = \widetilde O(\sqrt n)$.

翻译：对于离散乘积空间 $\prod_{i=1}^n \Omega_i'$ 上的分布，格劳伯动力学是一种马尔可夫链，其在每一步中，基于其他坐标的条件重置一个随机坐标。我们证明，$k$-格劳伯动力学（即重置一个包含 $k$ 个坐标的随机子集）在 $\chi^2$-散度上混合速度提升 $k$ 倍，并且在假设熵近似可张量化的条件下，在 KL-散度上混合速度也提升 $k$ 倍。我们将此应用于伊辛模型 $\mu_{J,h}(x)\propto \exp(\frac12\left\langle x,Jx \right\rangle + \langle h,x\rangle)$，其中 $\|J\|<1-c$（已知快速混合的区域）。我们证明，可以通过并行算法高效实现每一步 $\widetilde O(n/\|J\|_F)$-格劳伯动力学，从而得到运行时间为 $\widetilde O(\|J\|_F) = \widetilde O(\sqrt n)$ 的并行算法。