Closed-form survival probabilities for biased random walks at arbitrary step number

Abstract

We present a closed-form expression for the survival probability of a biased random walker to first reach a target site on a 1D lattice. The expression holds for any step number $N$ and is computationally faster than non-closed-form results in the literature. Because our result is exact even in the intermediate step number range, it serves as a tool to study convergence to the large $N$ limit. We also obtain a closed-form expression for the probability of last passage. In contrast to predictions of the large $N$ approximation, the new expression reveals a critical value of the bias beyond which the tail of the last-passage probability decays monotonically.