Random $3$-regular graphs are not globally synchronizing

Abstract

We study the homogeneous Kuramoto model on random 3-regular graphs. A connected graph $G$ is globally synchronizing if, for almost every initial condition, the gradient flow converges to a fully synchronized state, in which all phases coincide. It has been conjectured by Bandeira [Oberwolfach Rep. 21 (2024) 1163-1226] that a random 3-regular graph $G_{2n,3}$ is globally synchronizing with high probability. We disprove this conjecture in a strong way: $G_{2n,3}$ is not globally synchronizing with probability tending to 1.