As we know, the random walk on $\mathbb{Z}/n$ will converge(in some sense) to the Brownian motion on $\mathbb{R}$ when $n\to\infty$. I would like to know is there some higher dimensional analogy result.

**Edit:** As pointed by Nate Eldredge, there is a generalization of Donsker's theorem on manifold. But, I am interested in the following more topological generalization.

For a compact Riemannian manifold $X$, if a triangulation is given. Is there a canoncal way to defined a random walk $W$ on the vertex of the triangulation, such that $W^n$ the random walk defined in this way after $n$-th barycentric subdivision will converge to the Brownian motion on $X$.