Two lovers in a discrete place

[BMAD]

Somewhere there exists another world where the time-space fabric is discrete. We don't know the space fabric it might be 1D, 2D, 3D, or the space might be curved. Two lover breaks at zero point at zero time then start random walk. What is the probability they will they run into each other again?

[m00li]

Has to be zero

At time t1, they are equally likely to be at distance d apart or distance d1 apart or distance d2 apart or distance 0 apart as their velocities are not given and can be assumed to be anything. Similar distribution holds true for time t2 as well. So, there are infinite equally likely possibilities, and running into each other again is but one of them.

[plasmid]

I think the answer depends on the dimensionality of the space they're in.

People smarter than me have shown that a random walk will return to the origin with probability 1 in a 1-dimensional space, and probability 0 in high dimensional spaces.

[bonanova]

I think the answer depends on the dimensionality of the space they're in.

People smarter than me have shown that a random walk will return to the origin with probability 1 in a 1-dimensional space, and probability 0 in high dimensional spaces.

I was pondering the case for a single dimension.

With probability 1 they would each come back to the origin.

But the OP asks, essentially, for the first to return to WAIT there for the other to arrive.

Now we know just as there in no crying in baseball there is no waiting in random walks.

So what needs to be assessed is the likelihood of two chance events happening simultaneously.

I don't have an answer other than to guess it's zero.

But that's a ho-hum result, so maybe it's, surprisingly, unity.

I'm thinking the answer is either 1 or 0.

I can't imagine that it would be 0.25, for example.

I was unaware that p(return) in higher spaces was zero.

But that might lead to a final thought on the matter.

Perhaps two random walks in a single dimension behave as a single random walk in two dimensions.

The the answer of zero probability might then be surprising enough to be interesting.

[plasmid]

In a single dimension, the chance that they meet is 1.

Lover A will come back to the origin at some time T_A1 and, because he/she is now starting a new random walk from the origin, will return again at some time T_A2, and T_A3, and T_An. B will likewise return to the origin on an infinite number of occasions.

If A leaves the origin at time T_Ax heading in the positive direction on the number line, and B returns to the origin at time T_Ax+t before A returns to the origin again, and B came from the positive end of the number line, then their paths must have crossed. Because both A and B return to the origin an infinite number of times, that should eventually happen.

Actually, that argument might only be valid if 1) they each take steps of length = one Plank constant or 2) they are considered to have met if their paths crossed.