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 $n$.
Citation
P. Revesz. "Clusters of a Random Walk on the Plane." Ann. Probab. 21 (1) 318 - 328, January, 1993. https://doi.org/10.1214/aop/1176989406
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