Two observations on the capacity of the range of simple random walks on \(\mathbb{Z} ^3\) and \(\mathbb{Z} ^4\). (English) Zbl 1370.60041

Chang, Yinshan

Electron. Commun. Probab. 22, Paper No. 25, 9 p. (2017).

Summary: 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 A. Asselah et al. in [“Capacity of the range of random walk on \(\mathbb{Z}^4\)”, Preprint, arXiv:1611.04567].

MSC:

60F05	Central limit and other weak theorems
60G50	Sums of independent random variables; random walks
60J65	Brownian motion
60J10	Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords:

simple random walks; range; capacity; convergence in distribution; Brownian motion