riemannian geometry – Does a random walk on a surface visit uniformly?

Let $S$ be a smooth compact closed surface embedded in $mathbb{R}^3$ of genus $g$.
Starting from a point $p$, define a random walk as taking discrete steps
in a uniformly random direction,
each step a geodesic segment of the same length $delta$.
Assume $delta$ is less than the injectivity radius and small with respect to the intrinsic diameter of $S$.

Q. Is the set of footprints of the random walk evenly distributed on $S$,
in the limit? By evenly distributed I mean the density of points per unit area of surface
is the same everywhere on $S$.

This is likely known, but I’m not finding it in the literature on random walks
on manifolds. I’m especially interested in genus $0$.