Extracting subsets maximizing capacity and Folding of Random Walks

We prove that in any finite set of $mathbb Z^d$ with $dge 3$, there is a subset whose capacity and volume are both of the same order as the capacity of the initial set. As an application we obtain estimates on the probability of {it covering uniformly} a finite set, and characterize some {it folding} events, under optimal hypotheses. For instance, knowing that a region of space has an {it atypically high occupation density} by some random walk, we show that this random region is most likely ball-like