(In what follows, "cubic lattice" is short for "simple cubic lattice")
In this book, chapter 12, the following expression is derived for the probability distribution of a random walk on a $D-$dimensional cubic lattice (hypercubic lattice):
$$P_D(\vec x, t) = \left( \frac{D}{2 \pi t}\right)^{D/2} \exp \left(-\frac{D \|\vec x\|^2 }{2 t a^2}\right)$$
where $a$ is the lattice spacing and $\| \cdot\|$ is the euclidean norm.
I am interested in the equivalent expression in the case in which a non-hypercubic lattice is considered; in particular, I am interested in the $D=3$ case.
For example, it would be interesting to consider a diamond cubic or fcc lattice in $3D$, where the number of nearest neighbors is respectively $4$ and $12$, as opposed to the $6$ nearest neighbors of the cubic lattice.
I tried to work it out myself, but I'm having some trouble. So, the question is: is there any known (analytical or numerical) result for the three dimensional random walk on non-cubic lattice?
As I said, I am mostly interested in the $3D$ case, but every information regarding the $D-$dimensional case will be also much appreciated.
