A global synchronization theorem for oscillators on a random graph

Consider $n$ identical Kuramoto oscillators on a random graph. Specifically, consider Erdős–Rényi random graphs in which any two oscillators are bidirectionally coupled with unit strength, independently and at random, with probability $0≤p≤1$⁠. 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, $p∼log⁡(n)/n$ for $n≫1$⁠. Ling, Xu, and Bandeira made the first progress toward proving a result in this direction: they showed that if $p≫log⁡(n)/n1/3$⁠, then Erdős–Rényi networks of Kuramoto oscillators are globally synchronizing with high probability as $n→∞$⁠. Here, we improve that result by showing that $p≫log2⁡(n)/n$ suffices. Our estimates are explicit: for example, we can say that there is more than a $99.9996%$ chance that a random network with $n=106$ and $p>0.01117$ is globally synchronizing.