(a) Taking the front speed as the order parameter, there is a phase transition at a critical obstacle density depending on system size . For an infinite system, . For , we have . (b) Roughness of fully developed interfaces at for varying obstacle densities, as a function of the window length , such that . At , we find , consistent with the KPZ universality class. At , we find , consistent with the QEW universality class (see Theory in the Materials and methods section), which is known to be characterized by two roughness exponents, a local exponent (, panel b), and a global exponent when the roughness is measured over the whole system size (see panel c) (Amaral et al., 1995). Importantly for , there is a crossover length scale below which the system displays QEW scaling, whereas it returns to the KPZ scaling on longer length scales. (d) The time evolution of the interface also follows dynamics consistent with KPZ () and QEW () in the limiting cases. For intermediate , there is a crossover from qEW at short times to KPZ dynamics at longer times, before the roughness saturates at a -dependent value. For easier analysis, all simulations were performed in a box-like geometry.