Nonequilibrium defect-unbinding transition: Defect trajectories and loop statistics

In a Ginzburg-Landau model for parametrically driven waves, a transition between a state of ordered and one of disordered spatiotemporal defect chaos is found. To get insight into the breakdown of the order, the defect trajectories are tracked in detail. Since the defects are created and annihilated in pairs, the trajectories form loops in space-time. The probability distribution functions for the size of the loops and the number of defects involved in them undergo a transition from exponential decay in the ordered regime to a power-law decay in the disordered regime. These power laws are also found in a simple lattice model of randomly created defect pairs that diffuse and annihilate upon collision.