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 $t_{\min }\sim r^{1.33},$ which is different from the scaling of the most probable traveling time, $\tilde{t}\sim r^{1.64}.$ We also calculate the length of the path corresponding to the minimal traveling time and find ${\mathcal{l}}_{\min }\sim r^{1.13}$ and that the most probable traveling length scales as $\mathcal{l}̃\sim r^{1.21}.$ We present the relevant distribution functions and scaling relations.