First-passage-time distribution in a random random walk

A detailed study of the statistical properties of first-passage times in a one-dimensional random environment is presented. It is found that the spectrum of relaxation times of the master equation describing the hopping dynamics in such a system possesses a gap, which separates the leading relaxation time from the next one by an amount that is exponentially large in the square root of the size of the system. The leading relaxation time for a given (typical) system equals {τ}, the mean first-passage time for that realization. One consequence of the existence of this gap is that for times that are larger than a fraction of the mean first-passage time {τ}, the probability of the first-passage time to have a value t is approximated, to a high degree of accuracy, by (1/{τ})exp(-t/{τ}). The appearance of the above-mentioned gap is, surprisingly, related to the existence of a dominant single effective trap (or ``potential well''), the next leading trap being of no importance for large systems. A detailed study of the properties of the mean first-passage time and its fluctuations among the members of the ensemble of realizations is presented (the probability distribution of {τ} is shown to possess anomalous long-time behavior). The realization-averaged mean first-passage time is shown to be dominated by rare realizations and is contrasted with typical values of that quantity. The mean first-passage-time distribution is shown to be related to a new characterization of random walks that we coin the extent. The latter quantity is investigated in detail.