The cover time of a graph is the expected number of steps needed for the random walk to visit every node of the graph at least once. (We did not specify a start vertex. In general, one uses the start vertex from which the random walk talks the longest expecetd time to cover.)

The discrete torus modulo n is the graph on vertex set $\{1,2,...,n\}^2$ with an edge between (a,b) and (c,d) whenever (a-c, b-d) is congruent mod n to one of (1,0), (0,1), (-1,0), (0,-1). The cover time of this graph is asymptotic to $4 \pi n^2 \log n$. This was conjectured by Aldous (1989) and proved in a paper of Dembo, Peres, Rosen, and Zeitouni. The proof relies on the second moment method and uses (in a sense) the fact that the second moment method works for percolation on d-regular complete trees (like we discussed in class). The video below shows random walk covering the torus for n=64.