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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...)

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"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

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