Mean first-passage and residence times of random walks on asymmetric disordered chains

The problems of the first-passage time (FPT) [1, 2] and the residence time (RT) [3] are very important issues in random walk theory. Moreover, several properties of diffusion and transport in disordered systems are based in this concepts. Here, we do not want to present a survey of the enormous literature in the field of mean firstpassage time (MFPT) and mean residence time (MRT) of random walks, nevertheless, we wish to single out the works involved with analytical or exact results, mainly in one-dimensional disordered systems. Goldhirsch and Gefen [4] developed an analytical method for calculating MFPT for branched networks such as finite segments with dangling bonds and loops. The method is based on the generating function and was generalized for biased walks [5]. Extensions of the generating function method were done for analyzing the probability distribution function of FPT [6] and the current autocorrelation function [7]. Later on, the generating function method was used for random one-dimensional chains [8, 9], particularly for the Sinai problem [10]. Explicit expressions for the MFPT, in terms of the basic jump probabilities for discrete time random walk with a reflecting boundary were obtained independently by Van den Broeck [11], Le Doussal [12], and Murthy and Kehr [13] by different methods. It is interesting to remark that Gardiner [14] had previously reported explicit MFPT formulae. An exact solution of the generating function for the first-passage probability was presented by Raykin [15] using enumerative conbinatorics for summing up over