  \emph{Sampling} constitutes an important tool in a variety of areas: from
machine learning and combinatorial optimization to computational physics and
biology. A central class of sampling algorithms is the \emph{Markov Chain Monte
Carlo} method, based on the construction of a Markov chain with the desired
sampling distribution as its stationary distribution. Many of the traditional
Markov chains, such as the \emph{Glauber dynamics}, do not scale well with
increasing dimension. To address this shortcoming, we propose a simple local
update rule based on the Glauber dynamics that leads to efficient parallel and
distributed algorithms for sampling from Gibbs distributions.
  Concretely, we present a Markov chain that mixes in $O(\log n)$ rounds when
Dobrushin's condition for the Gibbs distribution is satisfied. This improves
over the \emph{LubyGlauber} algorithm by Feng, Sun, and Yin [PODC'17], which
needs $O(\Delta \log n)$ rounds, and their \emph{LocalMetropolis} algorithm,
which converges in $O(\log n)$ rounds but requires a considerably stronger
mixing condition. Here, $n$ denotes the number of nodes in the graphical model
inducing the Gibbs distribution, and $\Delta$ its maximum degree. In
particular, our method can sample a uniform proper coloring with $\alpha
\Delta$ colors in $O(\log n)$ rounds for any $\alpha>2$, which almost matches
the threshold of the sequential Glauber dynamics and improves on the $\alpha>2
+\sqrt{2}$ threshold of Feng et al.
