Partial and random covering times in one dimension.

We study the recently introduced random walk problems of partial covering time (PCT) and random covering time (RCT). We generalize the concept of first-passage time to a given set of m sites by considering the probability of visiting all m sites for the first time on the tth step. For the one-dimensional case we derive an explicit result for the mean time needed to visit m sites for the first time. Using this result we are able to solve the PCT and RCT problems exactly in one dimension.