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

TL;DR

This paper derives a closed-form, exact expression for the survival probability of a biased 1D random walk at any step number, enabling faster computation and insights into convergence and bias effects.

Contribution

It introduces a novel closed-form formula for survival and last passage probabilities in biased random walks, valid for any step number, improving computational efficiency and understanding of bias influence.

Findings

Exact survival probability formula valid for all steps.

Identification of a critical bias value affecting tail decay.

Enhanced understanding of convergence to large N behavior.

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.