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

Jacek Wszoła

TL;DR

The paper studies the first passage location $X(\tau_a)$ for a two-dimensional diagonal random walk on $\mathbb{Z}^2$ with transition probabilities depending on the current level $Y_n$, proving that the appropriately rescaled location is distributed in a bell-shaped manner. The approach combines generating-function techniques, recurrences, and the introduction of two new classes of rational functions, $\mathscr{P}$ and $\mathscr{Q}$, to obtain a product/continued-fraction representation that yields a convolution of geometric, mirrored-geometric, two-point, and $\mathscr{A}\mathscr{M}-\mathscr{C}\mathscr{M}$ components. The main result extends Bondesson-type bell-shaped phenomena to 2D Markov-additive processes and is shown to hold for standard $\mathbb{Z}^2$ walks and for honeycomb lattices via lattice embeddings. The findings provide a detailed structural description of the first-passage-location law and establish a rigorous framework for analyzing exit problems in fully discrete MAPs with barrier variants and multiple lattices, complemented by explicit examples.

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 $X(τ_a)$, where $τ_a$ is the first time $Y_n$ hits level $a \in \mathbb{Z}$. We prove that the probability mass function of appropriately rescaled $X(τ_a)$ is a convolution of geometric sequences, two-point sequences and an $\mathscr{AM}$-$\mathscr{CM}$ (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.

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

TL;DR

The paper studies the first passage location for a two-dimensional diagonal random walk on with transition probabilities depending on the current level , proving that the appropriately rescaled location is distributed in a bell-shaped manner. The approach combines generating-function techniques, recurrences, and the introduction of two new classes of rational functions, and , to obtain a product/continued-fraction representation that yields a convolution of geometric, mirrored-geometric, two-point, and components. The main result extends Bondesson-type bell-shaped phenomena to 2D Markov-additive processes and is shown to hold for standard walks and for honeycomb lattices via lattice embeddings. The findings provide a detailed structural description of the first-passage-location law and establish a rigorous framework for analyzing exit problems in fully discrete MAPs with barrier variants and multiple lattices, complemented by explicit examples.

Abstract

Let be a two-dimensional diagonal random walk on the lattice , with transition probabilities depending only on the position of . In this paper, we study its first passage locations , where is the first time hits level . We prove that the probability mass function of appropriately rescaled is a convolution of geometric sequences, two-point sequences and an - (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 and random walks on the honeycomb lattice.
Paper Structure (20 sections, 15 theorems, 132 equations, 5 figures)

This paper contains 20 sections, 15 theorems, 132 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Random walks and passage times
  3. Organization
  4. Generating functions of first passage locations
  5. Assumptions and notation
  6. Generating functions
  7. Recurrences
  8. Bell-shaped distributions
  9. Bell-shaped sequences
  10. Characterization theorems
  11. Rational functions
  12. Notation and definitions
  13. Properties of rational functions
  14. Proof of the main result
  15. Distributions of the first passage locations
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $(X_n, Y_n)$ be a diagonal random walk that starts in $(x,y) \in \mathbb{Z}^2$, with transition probabilities depending only on $Y_n$. For every $a \in \mathbb{Z}$, $a \neq y$, the first passage location $\frac{1}{2}(X(\tau_a)-|y-a|)-x$ has a bell-shaped probability mass function, given by the c where $0 \leqslant A \leqslant \lfloor \frac{1}{2}(|y-a|-1) \rfloor$ and $0 \leqslant B \leqslant \

Figures (5)

  • Figure 1: Sample trajectory of a two-dimensional symmetric diagonal random walk $(X_n, Y_n)$ starting from $(0,20)$. Here $X(\tau_0)=248$.
  • Figure 2: Embedding the honeycombs into the $\mathbb{Z}^2$ lattice.
  • Figure 3: Bell-shaped distribution of $\frac{1}{2}(X(\tau_0)-y)$ for a symmetric two-dimensional diagonal random walk on the lattice $\mathbb{Z}^2$ starting from $(0,y)$.
  • Figure 4: Bell-shaped distribution of $\frac{1}{2}(\tau_0-y)$ for one-dimensional random walk starting from $y$ with $q=1/4$.
  • Figure 5: Bell-shaped distribution of $\frac{1}{2}(X(\tau_0)-y)$ for a symmetric two-dimensional standard random walk on the lattice $\mathbb{Z}^2$ starting from $(0,y)$.

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 2.1
  • proof
  • Definition 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Definition 4.1
  • ...and 24 more