The Annals of Probability

Capacity of the range of random walk on $\mathbb{Z}^{4}$

Amine Asselah, Bruno Schapira, and Perla Sousi

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We study the scaling limit of the capacity of the range of a random walk on the integer lattice in dimension four. We establish a strong law of large numbers and a central limit theorem with a non-Gaussian limit. The asymptotic behaviour is analogous to that found by Le Gall in ’86 [Comm. Math. Phys. 104 (1986) 471–507] for the volume of the range in dimension two.

Article information

Ann. Probab., Volume 47, Number 3 (2019), 1447-1497.

Received: January 2017
Revised: March 2018
First available in Project Euclid: 2 May 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60G50: Sums of independent random variables; random walks

Capacity Green kernel law of large numbers central limit theorem


Asselah, Amine; Schapira, Bruno; Sousi, Perla. Capacity of the range of random walk on $\mathbb{Z}^{4}$. Ann. Probab. 47 (2019), no. 3, 1447--1497. doi:10.1214/18-AOP1288.

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