First passage locations for two-dimensional lattice random walks and the bell-shape

2025/01/24 by Jacek Wszoła, Wszoła, Jacek
Biochemistry, Genetics and Molecular Biology · Mathematics · #Diffusion and Search Dynamics #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2501.14393

openalex publication_date 2025/01/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (X_n, Y_n) be a two-dimensional diagonal random walk on the lattice ℤ², with transition probabilities depending only on the position of Y_n. In this paper, we study its first passage locations X(τ_a), where τ_a is the first time Y_n hits level a ∈ ℤ. We prove that the probability mass function of appropriately rescaled X(τ_a) is a convolution of geometric sequences, two-point sequences and an \mathscrAM-\mathscrCM (absolutely monotone then completely monotone) sequence. In particular, rescaled first passage locations have bell-shaped distributions. In order to prove our results, we introduce and study two new classes of rational functions with alternating zeros or poles. We also prove analogous theorems for standard random walks on the lattice ℤ² and random walks on the honeycomb lattice.

Related

• First passage statistics of Poisson random walks on lattices
• Random walks and Brownian motion: A method of computation for first-passage times and related quantities in confined geometries
• Topology of the support of the two-dimensional random walk
• Transient random walks on 2d-oriented lattices
• Visible lattice points in higher dimensional random walks and biases among them