I am wondering can we say something about the cover time $T$ for a box, eg. $[-n,n]^2\cap \mathbb{Z}^d$, by the simple symmetric random walk on $\mathbb{Z}^2$ starting from zero?


For example, the expected time, the variance, or some scale $a\_n$ such that $T\leq a\_n$ with high probability?


(I tried search on web but found nothing. Maybe I am not searching with the correct things, but...)



Cover time of a box by SRW in $\mathbb{Z}^2$?
"Cover times for Brownian motion
and random walks in two dimensions": Annals Math 160 (2004), 433-464,
By Dembo, Peres, Rosen, Zeitouni.


See Theorem 1.4


