On Covering paths with 3 Dimensional Random Walk

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: $\forall \epsilon >0,\exists c_{\epsilon }>0,$ $$P\left(\mathrm{Trace}(\mathcal{P})\subseteq \mathrm{Trace}(\{X_n{\}}_{n=0}^{\infty })\right)\le \exp \left(-c_{\epsilon }N{\log }^{-(1+\epsilon )}(N)\right).$$

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