Synchronization in disordered oscillator lattices: Nonequilibrium phase transition for driven-dissipative bosons

We show that lattices of phase oscillators with random natural frequencies undergo a transition from a desynchronized to a synchronized state for dimensions d < 4. The oscillators are described by a generalization of the nearest-neighbor Kuramoto model with an additional cosine coupling term. This model may be derived from the complex Ginzburg-Landau equations for a lattice of driven-dissipative Bose-Einstein condensates of exciton polaritons. We derive phase diagrams that classify the desynchronized and synchronized states, focusing on the behavior in one and two dimensions. This is achieved by outlining the connection of the oscillator model to the quantum description of localization of a particle in a random potential through a mapping to a modified Kardar-Parisi-Zhang equation. Our results indicate that synchronization in coupled polariton condensates and other examples of low-dimensional lattices of coupled oscillators is not destroyed by randomness in their natural frequencies.