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2018 On covering paths with 3 dimensional random walk
Eviatar B. Procaccia, Yuan Zhang
Electron. Commun. Probab. 23: 1-11 (2018). 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: $\forall \varepsilon >0,\exists c_\varepsilon >0,$ \[P\left ({\rm Trace}(\mathcal{P} )\subseteq{\rm Trace} \big (\{X_n\}_{n=0}^\infty \big ) \right )\le \exp \left (-c_\varepsilon N\log ^{-(1+\varepsilon )}(N)\right ).\]

Citation

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Eviatar B. Procaccia. Yuan Zhang. "On covering paths with 3 dimensional random walk." Electron. Commun. Probab. 23 1 - 11, 2018. https://doi.org/10.1214/18-ECP160

Information

Received: 25 September 2017; Accepted: 3 August 2018; Published: 2018
First available in Project Euclid: 12 September 2018

zbMATH: 1402.60053
MathSciNet: MR3863913
Digital Object Identifier: 10.1214/18-ECP160

Subjects:
Primary: 60G50, 60J10

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