  We study the problem of sampling an approximately uniformly random satisfying
assignment for atomic constraint satisfaction problems i.e. where each
constraint is violated by only one assignment to its variables. Let $p$ denote
the maximum probability of violation of any constraint and let $\Delta$ denote
the maximum degree of the line graph of the constraints.
  Our main result is a nearly-linear (in the number of variables) time
algorithm for this problem, which is valid in a Lov\'asz local lemma type
regime that is considerably less restrictive compared to previous works. In
particular, we provide sampling algorithms for the uniform distribution on:
  (1) $q$-colorings of $k$-uniform hypergraphs with $\Delta \lesssim q^{(k-4)/3
+ o_{q}(1)}.$
  The exponent $1/3$ improves the previously best-known $1/7$ in the case $q,
\Delta = O(1)$ [Jain, Pham, Vuong; arXiv, 2020] and $1/9$ in the general case
[Feng, He, Yin; STOC 2021].
  (2) Satisfying assignments of Boolean $k$-CNF formulas with $\Delta \lesssim
2^{k/5.741}.$
  The constant $5.741$ in the exponent improves the previously best-known $7$
in the case $k = O(1)$ [Jain, Pham, Vuong; arXiv, 2020] and $13$ in the general
case [Feng, He, Yin; STOC 2021].
  (3) Satisfying assignments of general atomic constraint satisfaction problems
with $p\cdot \Delta^{7.043} \lesssim 1.$
  The constant $7.043$ improves upon the previously best-known constant of
$350$ [Feng, He, Yin; STOC 2021].
  At the heart of our analysis is a novel information-percolation type argument
for showing the rapid mixing of the Glauber dynamics for a carefully
constructed projection of the uniform distribution on satisfying assignments.
Notably, there is no natural partial order on the space, and we believe that
the techniques developed for the analysis may be of independent interest.
