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] of Random Walks which gave upper bounds of $O(|V|^2)$ for various graphs, but none seem to apply to the grid case.