First encounters

摘要

A walker sets out on a random walk, moving between many different sites in no particular order. Formally, knowing the probability of arriving at each of the sites for every possible number of steps provides all the information required to characterize the walk. But it doesn't necessarily give easy answers to some relevant questions: for instance, what is the probability that the walker will reach a certain site for the first time after a given number of steps? This 'first-passage time' problem is notoriously intractable. For that reason, we often settle for the somewhat simpler problem of calculating the average (mean) first-passage time to an arbitrary site.