  The Lov\'asz Local Lemma is a classic result in probability theory that is
often used to prove the existence of combinatorial objects via the
probabilistic method. In its simplest form, it states that if we have $n$ `bad
events', each of which occurs with probability at most $p$ and is independent
of all but $d$ other events, then under certain criteria on $p$ and $d$, all of
the bad events can be avoided with positive probability.
  While the original proof was existential, there has been much study on the
algorithmic Lov\'asz Local Lemma: that is, designing an algorithm which finds
an assignment of the underlying random variables such that all the bad events
are indeed avoided. Notably, the celebrated result of Moser and Tardos [JACM
'10] also implied an efficient distributed algorithm for the problem, running
in $O(\log^2 n)$ rounds. For instances with low $d$, this was improved to
$O(d^2+\log^{O(1)}\log n)$ by Fischer and Ghaffari [DISC '17], a result that
has proven highly important in distributed complexity theory (Chang and Pettie
[SICOMP '19]).
  We give an improved algorithm for the Lov\'asz Local Lemma, providing a
trade-off between the strength of the criterion relating $p$ and $d$, and the
distributed round complexity. In particular, in the same regime as Fischer and
Ghaffari's algorithm, we improve the round complexity to $O(\frac{d}{\log
d}+\log^{O(1)}\log n)$. At the other end of the trade-off, we obtain a
$\log^{O(1)}\log n$ round complexity for a substantially wider regime than
previously known.
  As our main application, we also give the first $\log^{O(1)}\log n$-round
distributed algorithm for the problem of $\Delta+o(\Delta)$-edge coloring a
graph of maximum degree $\Delta$. This is an almost exponential improvement
over previous results: no prior $\log^{o(1)} n$-round algorithm was known even
for $2\Delta-2$-edge coloring.
