Renormalization of Oscillator Lattices with Disorder

A real-space renormalization transformation is constructed for lattices of non-identical oscillators with dynamics of the general form $d\phi_{k}/dt=\omega_{k}+g\sum_{l}f_{lk}(\phi_{l},\phi_{k})$. The transformation acts on ensembles of such lattices. Critical properties corresponding to a second order phase transition towards 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 $f_{lk}(\phi_{l},\phi_{k})=\phi(\phi_{l}-\phi_{k})$, where $\phi(x)$ is non-odd. The other model is pulse-coupled, with $f_{lk}(\phi_{l},\phi_{k})=\delta(\phi_{l})\phi(\phi_{k})$. Lower bounds for the critical dimensions for different types of coupling are obtained. For non-odd coupling, macroscopic synchronization cannot be ruled out for any dimension $D\geq 1$, whereas in the case of odd coupling, the well-known result that it can be ruled out for $D< 3$ is regained.