I think this is pretty known but I cannot find the answers online.
There's a random walk on the integer lattice of $d$ dimensions, where $d=2,3$. I am interested in:
- What is the probability that after $n$ steps, the walk will return to the origin?
- How many times will I expect to hit the origin within $n$ steps?
The answers for $d=1$ are easy, but I don't know them for $2,3$. Thanks a lot.