Capacity of the range of random walk: The law of the iterated logarithm

Amir Dembo, Izumi Okada

We establish both the $\mathrm{limsup}$ and the $\mathrm{liminf}$ law of the iterated logarithm (LIL), for the capacity of the range of a simple random walk in any dimension $d\ge 3$. While for $d…