Corrections to Scaling for Watersheds, Optimal Path Cracks, and Bridge Lines

We study the corrections to scaling for the mass of the watershed, the bridge line, and the optimal path crack in two and three dimensions. We disclose that these models have numerically equivalent fractal dimensions and leading correction-to-scaling exponents. We conjecture all three models to possess the same fractal dimension, namely, $d_f=1.2168\pm 0.0005$ in 2D and $d_f=2.487\pm 0.003$ in 3D, and the same exponent of the leading correction, $\mathrm{\Omega }=0.9\pm 0.1$ and $\mathrm{\Omega }=1.0\pm 0.1$, respectively. The close relations between watersheds, optimal path cracks in the strong disorder limit, and bridge lines are further supported by either heuristic or exact arguments.