Clusters of a Random Walk on the Plane

Abstract

Let $r(n)$ be the radius of the largest disc covered by $S(1),\ldots, S(n)$, where $\{S(k); k = 1, 2,\ldots\}$ is the simple symmetric random walk on $Z^2$. The main result tells us that $r(n) \geq n^{1/50}$ a.s. for all but finitely many $n$.