Universality of Synchrony: Critical Behavior in a Discrete Model of Stochastic Phase-Coupled Oscillators

We present the simplest discrete model to date that leads to synchronization of stochastic phase-coupled oscillators. In the mean field limit, the model exhibits a Hopf bifurcation and global oscillatory behavior as coupling crosses a critical value. When coupling between units is strictly local, the model undergoes a continuous phase transition which we characterize numerically using finite-size scaling analysis. In particular, the onset of global synchrony is marked by signatures of the $XY$ universality class, including the appropriate classical exponents $\beta$ and $\nu$, a lower critical dimension $d_{\mathrm{lc}}=2$, and an upper critical dimension $d_{\mathrm{uc}}=4$.