  Consider $n$ identical Kuramoto oscillators on a random graph. Specifically,
consider \ER random graphs in which any two oscillators are bidirectionally
coupled with unit strength, independently and at random, with probability
$0\leq p\leq 1$. We say that a network is globally synchronizing if the
oscillators converge to the all-in-phase synchronous state for almost all
initial conditions. Is there a critical threshold for $p$ above which global
synchrony is extremely likely but below which it is extremely rare? It is
suspected that a critical threshold exists and is close to the so-called
connectivity threshold, namely, $p\sim \log(n)/n$ for $n \gg 1$. Ling, Xu, and
Bandeira made the first progress toward proving a result in this direction:
they showed that if $p\gg \log(n)/n^{1/3}$, then \ER networks of Kuramoto
oscillators are globally synchronizing with high probability as
$n\rightarrow\infty$. Here we improve that result by showing that $p\gg
\log^2(n)/n$ suffices. Our estimates are explicit: for example, we can say that
there is more than a $99.9996\%$ chance that a random network with $n = 10^6$
and $p>0.01117$ is globally synchronizing.
