First-order like phase transition induced by quenched coupling disorder

We investigate the collective dynamics of a population of X Y model-type oscillators, globally coupled via non-separable interactions that are randomly chosen from a positive or negative value and subject to thermal noise controlled by temperature T. We find that the system at T = 0 exhibits a discontinuous, first-order like phase transition from the incoherent to the fully coherent state; when thermal noise is present (T > 0), the transition from incoherence to the partial coherence is continuous and the critical threshold is now larger compared to the deterministic case (T = 0). We derive an exact formula for the critical transition from incoherent to coherent oscillations for the deterministic and stochastic case based on both stability analysis for finite oscillators as well as for the thermodynamic limit (N → ∞) based on a rigorous mean-field theory using graphons, valid for heterogeneous graph structures. Our theoretical results are supported by extensive numerical simulations. Remarkably, the synchronization threshold induced by the type of random coupling considered here is identical to the one found in studies, which consider uniform input or output strengths for each oscillator node [H. Hong and S. H. Strogatz, Phys. Rev. E 84(4), 046202 (2011); Phys. Rev. Lett. 106(5), 054102 (2011)], which suggests that these systems display a "universal"character for the onset of synchronization.