Traveling time and traveling length in critical percolation clusters

We study traveling time and traveling length for tracer dispersion in two-dimensional bond percolation, modeling flow by tracer particles driven by a pressure difference between two points separated by Euclidean distance r. We find that the minimal traveling time t(min) scales as t(min) approximately r(1.33), which is different from the scaling of the most probable traveling time, t* approximately r(1.64). We also calculate the length of the path corresponding to the minimal traveling time and find l(min) approximately r(1.13) and that the most probable traveling length scales as l* approximately r(1.21). We present the relevant distribution functions and scaling relations.