arbiter
JournalsAccount

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

Jacek Wszoła

Electronic Journal of Probability2026https://doi.org/10.1214/26-ejp1498article
ABDC A
Weight
0.37

Abstract

Let (Xn,Yn) be a two-dimensional diagonal random walk on the lattice Z2, with transition probabilities depending only on the position of Yn. In this paper, we study its first passage locations X(τa), where τa is the first time Yn hits level a∈Z. We prove that the probability mass function of appropriately rescaled X(τa) is a convolution of geometric sequences, two-point sequences and an AM−CM (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.

1 citation

Open via your library →

Cite this paper

https://doi.org/https://doi.org/10.1214/26-ejp1498

Or copy a formatted citation

@article{jacek2026,
  title        = {{First passage locations for two-dimensional lattice random walks and the bell-shape}},
  author       = {Jacek Wszoła},
  journal      = {Electronic Journal of Probability},
  year         = {2026},
  doi          = {https://doi.org/https://doi.org/10.1214/26-ejp1498},
}

Paste directly into BibTeX, Zotero, or your reference manager.

Flag this paper

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

Flags are reviewed by the Arbiter methodology team within 5 business days.

Evidence weight

0.37

Balanced mode · F 0.40 / M 0.15 / V 0.05 / R 0.40

F · citation impact0.16 × 0.4 = 0.06
M · momentum0.53 × 0.15 = 0.08
V · venue signal0.50 × 0.05 = 0.03
R · text relevance †0.50 × 0.4 = 0.20

† Text relevance is estimated at 0.50 on the detail page — for your query’s actual relevance score, open this paper from a search result.