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## Random Walks on Integer Lattices of 2-3 dimensions [closed]

Hello,

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:

1. What is the probability that after $n$ steps, the walk will return to the origin?
2. 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.

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A Google search will give you more detailed info than any answer here. – Alex R. Dec 17 2010 at 21:00
While this is an interesting question, the answer is well known. Try looking up the name "Polya". – S. Carnahan Dec 17 2010 at 21:02