Average versus Typical Mean First-Passage Time in a Random Random Walk

Random walk in a one-dimensional random medium of length $N$ is analyzed. It is rigorously shown that in most realizations of the medium, the mean first-passage time, $\overline{t}$, bears the following relation to $N$, for large $N:log\overline{t}\propto \sqrt{N}$. The average of $\sqrt{t}$ over the realizations of the medium, $〈t〉$, satisfies $log〈t〉\propto N$. Our formalism, though being exact, employs only elementary means and makes transparent the physics of the delay experienced by the random walker: It is due to the existence of subsegments in which the bias against motion towards the desired end is largest. Some implications of these results concerning the replica method are briefly discussed.