Line crossing problem for biased monotonic random walks in the plane

TL;DR

This paper analyzes the crossing probability of biased monotonic random walks in the plane, establishing conditions under which the walk almost surely crosses a given line.

Contribution

It provides a new probabilistic result characterizing when biased monotonic random walks will almost surely cross a specified line in the plane.

Findings

If β ≥ ⌈α⌉, the walk crosses the line with probability 1 for all d ≥ 0.

The crossing probability depends on the bias parameter β relative to the slope α.

The paper extends understanding of crossing behaviors in biased lattice paths.

Abstract

In this paper, we study the problem of finding the probability that the two-dimensional (biased) monotonic random walk crosses the line $y=\alpha x+d$, where $\alpha,d \geq 0$. A $\beta$-biased monotonic random walk moves from $(a,b)$ to $(a+1,b)$ or $(a,b+1)$ with probabilities $1/(\beta + 1)$ and $\beta/(\beta + 1)$, respectively. Among our results, we show that if $\beta \geq \lceil \alpha \rceil$, then the $\beta$-biased monotonic random walk, starting from the origin, crosses the line $y=\alpha x+d$ for all $d\geq 0$ with probability 1.