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](http://mathworld.wolfram.com/PolyasRandomWalkConstants.html). 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 



What is the probability that two random walkers will meet? 
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. 


