  Markov Chain Monte Carlo (MCMC) algorithms are a widely-used algorithmic tool
for sampling from high-dimensional distributions, a notable example is the
equilibirum distribution of graphical models. The Glauber dynamics, also known
as the Gibbs sampler, is the simplest example of an MCMC algorithm; the
transitions of the chain update the configuration at a randomly chosen
coordinate at each step. Several works have studied distributed versions of the
Glauber dynamics and we extend these efforts to a more general family of Markov
chains. An important combinatorial problem in the study of MCMC algorithms is
random colorings. Given a graph $G$ of maximum degree $\Delta$ and an integer
$k\geq\Delta+1$, the goal is to generate a random proper vertex $k$-coloring of
$G$.
  Jerrum (1995) proved that the Glauber dynamics has $O(n\log{n})$ mixing time
when $k>2\Delta$. Fischer and Ghaffari (2018), and independently Feng, Hayes,
and Yin (2018), presented a parallel and distributed version of the Glauber
dynamics which converges in $O(\log{n})$ rounds for $k>(2+\varepsilon)\Delta$
for any $\varepsilon>0$. We improve this result to $k>(11/6-\delta)\Delta$ for
a fixed $\delta>0$. This matches the state of the art for randomly sampling
colorings of general graphs in the sequential setting. Whereas previous works
focused on distributed variants of the Glauber dynamics, our work presents a
parallel and distributed version of the more general flip dynamics presented by
Vigoda (2000) (and refined by Chen, Delcourt, Moitra, Perarnau, and Postle
(2019)), which recolors local maximal two-colored components in each step.
