First passage times and distances along critical curves

Abstract

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.

5 pages, 5 figures

References in corpus (6)

• Critical Percolation in Finite Geometries
• SLE for theoretical physicists
• Hausdorff dimensions for SLE_6
• Stochastic geometry of critical curves, Schramm-Loewner evolutions, and conformal field theory
• Monte Carlo Tests of SLE Predictions for the 2D Self-Avoiding Walk
• Outflow probability for drift--diffusion dynamics

Cited by in corpus (5)

• First passage time processes and subordinated SLE
• Continuous-time random-walk approach to normal and anomalous reaction-diffusion processes
• Mass transport subject to time-dependent flow with non-uniform sorption in porous media
• A nonlinear random walk approach to concentration-dependent contaminant transport in porous media
• Pre-asymptotic corrections to fractional diffusion equations