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

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1493777159

Digital Object Identifier
doi:10.1214/17-ECP55

Mathematical Reviews number (MathSciNet)
MR3652038

Zentralblatt MATH identifier
1370.60041

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords
capacity range of simple random walks

Rights
Creative Commons Attribution 4.0 International License.

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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References

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