For a random walk in integer valued order pair coordinate, if a particle/agent moves randomly right/left/up/down one unit with equal probability, what is the ratio of unique coordinate visited to total coordinate visited? Is there any analytic way to find this? I use simulations to verify that it is non-zero. Does it converge to any value and what is its relation with dimension?
For example $A$ starts from $(0,0) \rightarrow (0,1) \rightarrow (1,1)\rightarrow (1,0)\rightarrow (1,1)\rightarrow (0,1)\rightarrow (-1,1)\rightarrow (-1,0)$ etc. In this case, $A$ visits $8$ locations but it repeats $2$ locations. So the ratio is $0.75$. If we continue this process, what is the limiting ratio?
In $\textit{Brownian}$ Motion $A$ visits zero infinitely often at the beginning, but how that would help to solve this problem!
This question was answered by Dvoretzky & Erdős. Specifically, the expected proportion of different sites visited in the first $n$ steps is given (asymptotically) by $$ \frac{\pi}{\ln n}+O\!\left(\frac{\ln\ln n}{\ln^2 n}\right). $$