# On Covering paths with 3 Dimensional Random Walk

## TL;DR

This paper investigates the probability that a 3D simple random walk covers a specific path, providing an upper bound that decays slightly slower than exponential as the path length increases.

## Contribution

It extends previous results by deriving a new upper bound for the covering probability in three dimensions, where earlier techniques for higher dimensions do not apply.

## Key findings

- Probability decays as N log^{-(1+ε)}(N) for 3D walks
- Upper bound is weaker than exponential decay in 3D
- Results differ from known behavior in higher dimensions

## Abstract

In this paper we find an upper bound for the probability that a $3$ dimensional simple random walk covers each point in a nearest neighbor path connecting 0 and the boundary of an $L_1$ ball of radius $N$. For $d\ge 4$, it has been shown in [5] that such probability decays exponentially with respect to $N$. For $d=3$, however, the same technique does not apply, and in this paper we obtain a slightly weaker upper bound: $\forall \varepsilon>0,\exists c_\varepsilon>0,$ $$P\left({\rm Trace}(\mathcal{P})\subseteq {\rm Trace}\big(\{X_n\}_{n=0}^\infty\big) \right)\le \exp\left(-c_\varepsilon N\log^{-(1+\varepsilon)}(N)\right).$$