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Claim. Let $T_r$ be the first time at which symmetric nearest-neighbor simple random walk on $\mathbb Z^2$, started at the origin, has visited every lattice point of the disc of radius $r$. P. Révész, Random Walk in Random and Non-Random Environments, World Scientific, Teaneck, 1990, proves that there are constants $0<a<b<\infty$ with

$$e^{-b/t}\le\liminf_{r\to\infty}\mathbb P\bigl(\log T_r\le t(\log r)^2\bigr) \le\limsup_{r\to\infty}\mathbb P\bigl(\log T_r\le t(\log r)^2\bigr) \le e^{-a/t}$$

for every $t>0$, and conjectures that the limit exists and has the form $e^{-\lambda/t}$. The statement is taken from the introduction of the 2004 paper of Dembo, Peres, Rosen and Zeitouni (printed p. 436, on its library card), which reports the bound as proved independently by Kesten and by Révész and cites the monograph for Révész's proof; the monograph's theorem number and constants are not recorded here. For the path-dependent radius $R_n$ of the largest origin-centered lattice disc covered by time $n$, the event $R_n\ge r$ is the event $T_r\le n$, so the bound says that $\log R_n/\sqrt{\log n}$ is bounded above and below in probability: $R_n$ has the typical scale $\exp(\sqrt{\log n})$ that the 1999 booklet's Problem 6.76 conjectured. The change of variables, with its moving thresholds and disc-boundary conventions, is the one the library's radius deduction carries out for the limit law. This is the corrected Statement of Problem 1164, the order of $\log R_n$ in probability. The bound gives neither a limit law nor its rate; the exponential limit with rate $4$ is the claim of Dembo, Peres, Rosen and Zeitouni.

Acceptance. Reviewed: Thomas Bloom, the curator of erdosproblems.com, labels the problem proved and credits the asymptotic as proved independently by Révész [Re90] and Kesten, and the refereed 2004 paper reports the same attribution. The monograph is a book, not a journal article, so refereed is not listed. Kesten's proof has no publication of his own: the 2004 paper cites it as quoted by Aldous and by Lawler, so it has no claim page and is disclosed here as the independent co-credit. Lawler later published the same bound with $a=2$ and $b=4$ (G. Lawler, On the covering time of a disc by a random walk in two dimensions, Seminar in Stochastic Processes 1992, Birkhäuser (1993), 189–208), as the 2004 introduction reports (printed p. 436); the site does not credit it, so it is disclosed here and has no claim page. The page is dated by the monograph's publication month, September 1990.

Depends on. Nothing in this wiki: the bound is the monograph's own.