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2017 Two observations on the capacity of the range of simple random walks on $\mathbb{Z} ^3$ and $\mathbb{Z} ^4$
Yinshan Chang
Electron. Commun. Probab. 22: 1-9 (2017). DOI: 10.1214/17-ECP55

Abstract

We prove a weak law of large numbers for the capacity of the range of simple random walks on $\mathbb{Z} ^{4}$. On $\mathbb{Z} ^{3}$, we show that the capacity, properly scaled, converges in distribution towards the corresponding quantity for three dimensional Brownian motion. The paper answers two of the three open questions raised by Asselah, Schapira and Sousi in [2, Section 6].

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Yinshan Chang. "Two observations on the capacity of the range of simple random walks on $\mathbb{Z} ^3$ and $\mathbb{Z} ^4$." Electron. Commun. Probab. 22 1 - 9, 2017. https://doi.org/10.1214/17-ECP55

Information

Received: 6 December 2016; Accepted: 26 April 2017; Published: 2017
First available in Project Euclid: 3 May 2017

zbMATH: 1370.60041
MathSciNet: MR3652038
Digital Object Identifier: 10.1214/17-ECP55

Subjects:
Primary: 60J10

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