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.
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> 
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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>



Reference Request: Cover time for simple random walk on $n \times n$ torus
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 $n^2(\log n)^2$.


