Reference Request: Cover time for simple random walk on $n \times n$ torus I'm putting the finishing touches on my masters thesis and need a reference for the following fact (which my advisor told me):

Let $G$ be the $n\times n$ grid and identify the sides to make it a torus. A simple random walk on $G$ is expected to take $O(n^4)$ time before it hits every vertex.

I've been googling for this all day with no luck. Does anyone know where I can find a statement of this result? I don't need the original reference...a textbook would be fine. Also, if you have a reference for $G$ being the $n\times n$ grid without sides identified I can make that work too. I just need this fact for the "previous work" section of the thesis...none of my results depend on it.
I read several things in [Lovasz's Survey][1] of Random Walks which gave upper bounds of $O(|V|^2)$ for various graphs, but none seem to apply to the grid case.
[1] http://www.cs.unibo.it/babaoglu/courses/cas/resources/tutorials/RandomWalks.pdf
 A: Markov Chains and Mixing Times by Levin, Peres and Wilmer. Section 11.3.2. 
http://research.microsoft.com/en-us/um/people/peres/markovmixing.pdf
The expected cover time is of order $n^2(\log n)^2$.
A: The asymptotic cover time result in [1]  (see comment 2 above)
works for the grid as well as the torus. See comment 2 on page 28 in https://arxiv.org/pdf/math/0107191.pdf . Further refinements are in  refs [2] and [3] below. In particular [2] considers the cover time of the 2d grid with a wired boundary. The boundary conditions do not affect the first order asymptotics.
[1] Dembo, Amir, Yuval Peres, Jay Rosen, and Ofer Zeitouni. "Cover times for Brownian motion and random walks in two dimensions." Annals of mathematics (2004)Vol. 160, No. 2 pp. 433-464.    
[2]  Ding, Jian. "On cover times for 2D lattices." Electron. J. Probab 17, no. 45 (2012): 1-18.  
[3] Belius, David, and Nicola Kistler. "The subleading order of two dimensional cover times." Probability Theory and Related Fields (2014): 1-92.
