The first positive position of a lattice random walk

2025/10/20 by Godrèche, Claude, Luck, Jean-Marc
#FOS: Mathematics #FOS: Physical sciences #Probability (math.PR) #Statistical Mechanics (cond-mat.stat-mech)

paper · doi:10.48550/arxiv.2510.17329

Abstract

The distribution of the first positive position reached by a random walker starting at the origin is central to the analysis of extremes and records in one-dimensional random walks. In this work, we present a detailed and self-contained analytical study of this distribution for symmetric finite-range lattice walks, whose steps are drawn from a distribution supported on finitely many integers.

Citations

• First-passage statistics of random walks: a general approach via Riemann-Hilbert problems
• New insights into the distribution of the topmost gap in random walks and Lévy flights
• On the first positive position of a random walker

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