Renormalization of oscillator lattices with disorder

A real-space renormalization transformation is constructed for lattices of nonidentical oscillators with dynamics of the general form dφₖ/dt=ωₖ+g∑ₗfₗₖ(φₗ,φₖ). The transformation acts on ensembles of such lattices. Critical properties corresponding to a second-order phase transition toward macroscopic synchronization are deduced. The analysis is potentially exact but relies in part on unproven assumptions. Numerically, second-order phase transitions with the predicted properties are observed as g increases in two structurally different two-dimensional oscillator models. One model has smooth coupling fₗₖ(φₗ,φₖ)=φ(φₗ-φₖ), where φ(x) is nonodd. The other model is pulse coupled, with fₗₖ(φₗ,φₖ)=δ(φₗ)φ(φₖ). Lower bounds for the critical dimensions for different types of coupling are obtained. For nonodd coupling, macroscopic synchronization cannot be ruled out for any dimension D≥1, whereas in the case of odd coupling, the well-known result that it can be ruled out for $D<3$ is regained.