It is a well known result that a random walk on a 2D lattice will return to the origin see Polya's random walk constant. Based on this, it is not a big stretch to conclude that the random walk will visit every point of the plane with probability 1. A bit more surprising is the fact that this is not true in higher dimensions (see the link above).
My question is the following: What is the probability that two random walks with distinct origins will arrive at the same point after the same number of steps?
I think it's pretty clear that the answer will depend on the distance between the origins of the walks. So far, I've been trying reduce this to a problem of one random walk in a higher dimensional lattice, but I'm not sure if this is a good approach.
In case the answer is obvious, this problem is easy generalize (higher dimensional lattices, more random walks, etc.). I'd appreciate a reference or two where I could learn more.
Thanks
Best Answer
The difference between the positions is another random walk in the same dimension. You can either view the steps as different, or sample a random walk at even times. So, the probability is $0$ if meeting is ruled out by parity, and $1$ in the plane if meeting is possible.