Please imagine a discrete random walk on an N-dimensional rectangular lattice with dimensional lengths $(l\_1, ..., l\_N) \in L$ and total lattice points $P = \prod{l\_i}$, for $i = 1, ..., N$. At each time step, the walker will move to one of it's adjacent lattice points with equal probability. The N-dimensional random walk is non-self-avoiding, the walker must move with each time step, and the boundaries of the lattice are reflecting. However, jump probabilities must be adjusted at edges and corners due to a reduction in the number of adjacent nodes - i.e. jump probabilities will vary from $\frac{1}{2N}$ internal to the lattice to $\frac{1}{N}$ at the edges of the lattice. 


Provided the random walk specifications above, what might be the expected step-time distribution for the walker visiting every position in the N-dimensional rectangular lattice with dimensional lengths $L$? 



Expected number of steps for a discrete random walk to visit every point on an N-dimensional rectangular lattice
If you look for "cover time of graph" you will find a lot of references, cf. e.g. "Jonasson, Schramm, ON THE COVER TIME OF PLANAR GRAPHS, Elect. Comm. in Probab. 5 (2000) 85-90, <http://www.emis.de/journals/EJP-ECP/_ejpecp/ECP/include/getdocbfb7.pdf>. In this paper you find the following result by Zuckermann (for details pls. cf the paper): 


If $G$ is a finite portion of a $d$-dimensional lattice $\mathbb{Z}$ with $n$ vertices, the expected cover time (to visit all vertices) is $\Theta(n^2)$ for $d=1$, $\Theta(n (\log n)^2)$ for $d=2$, and $\Theta(n \log n)$ for $d \ge 3$.


