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

Abstract

We establish both the lim sup and the lim inf law of the iterated logarithm (lil) for the capacity of the range of a simple random walk in any dimension $\mathit{d}\ge 3$. While for $\mathit{d}\ge 4$, the order of growth in n of such lil at dimension d matches that for the volume of the random walk range in dimension $\mathit{d}-2$, somewhat surprisingly this correspondence breaks down for the capacity of the range at $\mathit{d}=3$. We further establish such lil for the Brownian capacity of a three-dimensional Brownian sample path and novel, sharp moderate deviations bounds for the capacity of the range of a four-dimensional simple random walk.