Return probability on a lattice

To begin, we consider a basic example of a discrete first passage process. Consider an unbiased Bernoulli walk on the integers starting at the origin. We wish to determine the probability, R, that the walker eventually returns to 0, regardless of the number of steps it takes. Without loss of generality, say the walker’s first step is to 1. From here, the walker can either step left to 0 or can return to 1 n times before stepping back to 0. To avoid double counting, we must also assert that if the walker comes back to 1 n times, it cannot step left to 0 any of those times except the last. Since half of the possible paths from 1 back to 1 stay to the right of 1, the probability of returning to 1 exactly once before hitting 0 is R/2. Likewise, the probability of returning to 1 n times before stepping back to 0 is (R/2)n . Thus