Estimates of the largest disc covered by a random walk (Q2641008)

Let \((S_ n)\) be symmetric random walk in \({\mathbb{Z}}^ 2\) started from the origin and let \(Q(N)=\{x=(i,j):\| x\| =(i^ 2+j^ 2)^{1/2}\leq N\},\) \(N\in {\mathbb{N}}\). Say that Q(N) is covered by time n if \((S_ n)\) has visited all sites in Q(N) by time n. Let \(R(n)=\max \{N:\) \((S_ n)\) covers Q(N) by time \(n\}\). The author shows that for each \(\epsilon >0\), then a.s. \[ R(n)\geq \exp [(120)^{-1/2}(1-\epsilon)(\log n \log_ 3n)^{1/2}] \] for infinitely many n. This complements a known two-sided inequality, see e.g., \textit{P. Erdős} and author [J. Multivariate Anal. 27, No.1, 169-180 (1988; Zbl 0655.60055)]. Next it it shown that \[ \liminf_{n\to \infty}P\{(\log R(n))^ 2/\log n>z\}\geq \exp (-120z), \] thus completing an upper bound due to the author [Almost everywhere convergence, Proc. Int. Conf., Columbus/OH 1988, 369-392 (1989; Zbl 0687.60066)].