On covering paths with 3 dimensional random walk

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 L₁ ball of radius N in ℤ^d. For d ≥ 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: ∀ε > 0, ∃c_ε > 0, (Formula Presented).