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