First-passage times and distances along critical curves

We propose a model for anomalous transport in inhomogeneous environments, such as fractured rocks, in which particles move only along pre-existing self-similar curves (cracks). The stochastic Loewner equation is used to efficiently generate such curves with tunable fractal dimension d_f. We numerically compute the probability of first passage (in length or time) from one point on the edge of the semi-infinite plane to any point on the semi-circle of radius R. The scaled probability distributions have a variance which increases with d_f, a non-monotonic skewness, and tails that decay faster than a simple exponential. The latter is in sharp contrast to predictions based on fractional dynamics and provides an experimental signature for our model.