I'm imagining a $2D$ lattice, but insight into higher dimensional lattices would also be helpful. My question is related to the number of intersections of a random walk on lattice with itself. However, that is not my exact question.
In particular, I want to know the probability distribution describing the number of visited unique grid points after a number $t$ steps. I know that the late-time position, with a little blurring, can be well-described by a gaussian with standard deviation proportional to $\sqrt{t}$, yielding root-mean-square expectation values proportional to $\sqrt{t}$. What I want to know is the behavior of the number of visited grid points. For example, does the number of visited grid points grow like $t$ or $\sqrt{t}$ or some other power law? Ultimately, I want to know asymptotic behavior, like the probability of having visited just a few dozen grid points after a very long time as a function of $t$ and $n$.
To make clear what I mean by visited grid points, I have the following pictures. The steps are ordered from light to dark.
For example, in the drawing below, this specific instantiation of a random walk has visited 7 grid points after 7 steps, which I have colored a light pink. It started at the middle left and ended in the upper right.
Here's another realization of a random walk, this time reaching 5 grid points after 7 steps:
To summarize, my question is: What is the probability distribution describing the number of visited unique grid points $n$ after a number $t$ steps on a $2D$ lattice? I am particularly interested in $n/t$ small.
Edit: I should add that I anticipate an exponential decay in $t$ at large $t$ at least. The reason for this is that the equivalent $1D$ problem is closely related to the study of exit times from some fixed interval of width $n$, and the probability that an exit time is greater than some $t$ is bounded from above by an exponential decay in $t$: Exit Time of an Interval Brownian Motion - Distribution discusses this.