On covering paths with 3 dimensional random walk

Eviatar B Procaccia,Yuan Zhang

doi:10.1214/18-ecp160

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

Highlights

We study the probability that the trace of a nearest neighbor path in Z3 connecting 0 and the boundary of a L1 ball of radius N is completely covered by the trace of a 3 dimensional simple random walk
We say that a path P is covered by a d dimensional random walk {Xd,n}∞ n=0 if
In [5], we have shown that for any d ≥ 2 such covering probability is maximized over all nearest neighbor paths connecting 0 and ∂B1(0, N) by the monotonic path that stays within distance one above/below the diagonal x1 = x2 = · · · = xd

Summary

Introduction

We study the probability that the trace of a nearest neighbor path in Z3 connecting 0 and the boundary of a L1 ball of radius N is completely covered by the trace of a 3 dimensional simple random walk. (Theorem 1.5 in [5]) There is a Pd ∈ (0, 1) such that for any nearest neighbor path P = (P0, P1, · · · , PK) connecting 0 and ∂B1(0, N ) and {Xd,n}∞ n=0 which is a d−dimensional simple random walk starting at 0 with d ≥ 4, we always have P Trace(P) ⊆ Trace {Xd,n}∞ n=0 ≤ Pd[N/d].

Infinite Transient Subset of the Diagonal

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