Do Random Walks on the Hexagonal Lattice have a limit? For every positive integer $n$, consider a regular hexagon $\mathrm{H}_n$ such that 
the distance of each vertex from the center is $\frac{1}{\sqrt{n}}$. That in turn 
induces a tiling of $\mathbb{R}^2$. Let us call that tiling $\mathcal{T}_n$ 
(assume that one of the vertices is at the origin). 
For each such tiling consider the following random walk: Start from the origin 
at $t=0$. At $t= \frac{1}{n}$ there is a $\frac{1}{3}$ probability to move 
to any of the neighboring vertex; continue this $n$ times till $t=1$. Join these 
$n$ points by a straight line to get a continuous map from $[0,1]$ to $\mathbb{R}^2$. There are obviously $3^n$ different continuous paths this way. 
Notice that each $n$ gives us a probability measure $\mu_n$ on the space 
$$ \Omega:= \{ f:[0,1] \rightarrow \mathbb{R}^2: f(0) =0,  ~~~~f ~~\text{is continuous}\}.$$ 
(choose each of those walks with probability $\frac{1}{3^n}$, any other path 
with probability zero). 
My question is the following: Do these probability measures converge in a 
weak sense to some measure $\mu$ on $\Omega$? 
By weak convergence I mean that for any bounded continuous 
function $\Phi: \Omega \rightarrow \mathbb{R}$ 
$$ \int \Phi(f) d \mu_n \rightarrow \int \Phi(f) d\mu$$
Here $\Omega$ is to be thought of as a metric space with 
supremum norm.  
Note that, if the tiling was a square tiling with length $\frac{1}{\sqrt{n}}$, 
then these measures would converge. I believe this is one way to 
construct the standard Brownian Motion. 
 A: Yes, it converges to Brownian motion by Donsker's theorem. This is an example of what's known as universality in statistical mechanics: the large scale dynamics of a system should be independent of its microscopic geometry.
A: If you want to derive the Functional CLT for this process from Donsker's theorem, you have to add a couple of small ingredients. 
Donsker's applies to random walks with i.i.d. steps. Yours is not quite of that kind. Two consecutive steps are not i.i.d: (i)one imposes a direction restriction on another; (ii) also if you color the vertices of the lattice in two colors appropriately, even steps will go from black to white and odd steps will go from white to black. 
There are many ways to overcome this. One is to notice that if you observe your process only at even times, then the resulting process has independent steps, so you can apply Donsker's to it. It remains to estimate the difference between the new process and the original one, that is easy.
A more general way is to use the martingale problems approach. See the chapter of Ethier & Kurtz on diffusion approximation.
