Two and three dimensional random walks

The algorithms and methods explained in the previous sections can be easily generalized to more dimensions. Consider a discrete lattice, either rectangular or triangular. The difference is the coordination number $z$, the number of nearest neighbor sites. In the rectangular lattice, we can move to $z=4$ different sites, while in the triangular, we can move to $z=6$ different sites. We can define the ``root mean square displacement'' (rms) as

$$R_N\equiv \sqrt{\langle \Delta R_N^2}.$$