Anomalous Critical Behavior of Driven Disordered Systems Beyond the Overdamped Limit

Elastic disordered interfaces driven through a heterogenpeous landscape respond via intermittent avalanches. In the overdamped limit, avalanche sizes follow a scale-free power-law distribution. Here, we investigate how inertial and dynamical terms beyond the overdamped approximation modify this behavior, focusing on the combined effects of interaction range, dimensionality, and dissipation. Using large-scale numerical simulations, we show that these additional terms robustly generate system-spanning events-so-called kings-responsible for a characteristic bump in the avalanche size distribution, which arise only in the presence of dissipation. We identify two distinct mechanisms underlying this phenomenon. In two-dimensional systems with long-range elasticity, we observe genuine kings, which persist in the thermodynamic limit and are consistent with mean field predictions. In contrast, one-dimensional systems with long-range elasticity and systems with short-range interactions in both 1D and 2D exhibit ghost kings: failed synchronization events producing large, but finite, ballistically spreading avalanches. Our findings clarify the origin of bump features in avalanche statistics and establish the conditions under which true system-spanning events emerge in realistic, dissipative systems.