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

Debendro Mookerjee and Sarah Kostinski*

  • *Contact author: sk10775@nyu.edu
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Phys. Rev. Research 7, L042007 – Published 3 October, 2025

DOI: https://doi.org/10.1103/krxn-vqnt

Abstract

We present a closed-form expression for the survival probability of a biased random walker to first reach a target site on a one-dimensional 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.

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Phys. Rev. Research 7, L042007– Published 3 October, 2025
  • Received 30 May 2025
  • Accepted 27 August 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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