Capacity of the range of random walk on \(\mathbb{Z}^{4}\). (English) Zbl 1467.60017

Summary: 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 J. F. Le Gall [Commun. Math. Phys. 104, 471–507 (1986; Zbl 0609.60078)] for the volume of the range in dimension two.


60F05 Central limit and other weak theorems
60G50 Sums of independent random variables; random walks
60F15 Strong limit theorems


Zbl 0609.60078
Full Text: DOI arXiv Euclid


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