On Covering paths with 3 Dimensional Random Walk

2017/05/10 by Procaccia, Eviatar B., Zhang, Yuan
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1705.03915

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 L₁ ball of radius N. 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, P(\rm Trace(P)⊆ \rm Trace(\X_n\_n=0^∞) )≤ exp(-c_ε Nlog^-(1+ε)(N)).

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