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 can implement each step of $\widetilde \Theta(\sqrt n)$-Glauber dynamics efficiently and obtain running time $\widetilde O(\sqrt n)$.

翻译：对于定义在离散乘积空间 $\prod_{i=1}^n \Omega_i'$ 上的分布，格劳伯动力学是一种马尔可夫链，其在每一步根据其他坐标的条件分布重采样一个随机坐标。我们证明，$k$-格劳伯动力学（每次重采样一个包含 $k$ 个坐标的随机子集）在 $\chi^2$ 散度意义下的混合速度加快了 $k$ 倍，并且在熵近似张量化的假设下，其在 KL 散度意义下的混合速度也加快了 $k$ 倍。我们将此应用于两种场景下的并行算法设计：(1) 对于伊辛模型 $\mu_{J,h}(x)\propto \exp(\frac1 2\left\langle x,Jx \right\rangle + \langle h,x\rangle)$，在 $\|J\|<1-c$ 的条件下（即已知存在快速混合的区域），我们证明了可以高效地通过并行算法实现 $\widetilde \Theta(n/\|J\|_F)$-格劳伯动力学的每一步，从而得到一个运行时间为 $\widetilde O(\|J\|_F) = \widetilde O(\sqrt n)$ 的并行算法。(2) 对于足够高温下的混合 $p$-自旋模型，我们证明了以高概率可以高效地实现 $\widetilde \Theta(\sqrt n)$-格劳伯动力学的每一步，并获得 $\widetilde O(\sqrt n)$ 的运行时间。