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

Abstract

Let $(X_n, Y_n)$ be a two-dimensional diagonal random walk on the lattice $\mathbb{Z}^2$, with transition probabilities depending only on the position of $Y_n$. In this paper, we study its first passage locations $X(\tau_a)$, where $\tau_a$ is the first time $Y_n$ hits level $a \in \mathbb{Z}$. We prove that the probability mass function of appropriately rescaled $X(\tau_a)$ is a convolution of geometric sequences, two-point sequences and an $\mathscr{AM}$-$\mathscr{CM}$ (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 $\mathbb{Z}^2$ and random walks on the honeycomb lattice.