## Electronic Communications in Probability

### Two observations on the capacity of the range of simple random walks on $\mathbb{Z} ^3$ and $\mathbb{Z} ^4$

Yinshan Chang

#### 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].

#### Article information

Source
Electron. Commun. Probab., Volume 22 (2017), paper no. 25, 9 pp.

Dates
Accepted: 26 April 2017
First available in Project Euclid: 3 May 2017

https://projecteuclid.org/euclid.ecp/1493777159

Digital Object Identifier
doi:10.1214/17-ECP55

Mathematical Reviews number (MathSciNet)
MR3652038

Zentralblatt MATH identifier
1370.60041

#### Citation

Chang, Yinshan. Two observations on the capacity of the range of simple random walks on $\mathbb{Z} ^3$ and $\mathbb{Z} ^4$. Electron. Commun. Probab. 22 (2017), paper no. 25, 9 pp. doi:10.1214/17-ECP55. https://projecteuclid.org/euclid.ecp/1493777159

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