I was wondering about a question about Random Walks. I came across various papers where the probability of 2 random walkers in 1 dimension and 2 dimension starting at the same point and returning to the origin was found. I was doing a research on a question similar to this - What is the probability that 2 random walkers, originating from the same point, meet at a point other than the origin?
I tried solving the cases for 1 dimension and 2 dimension but I was stuck because the displacement does not become 0 in this case. I couldn't handle the displacement term. I would be glad if someone could give me some hint or tell me how I should proceed further.
Assuming $1D$ random walk:
For the two random walkers($W_1, W_2$) to meet at some time point $N$, the number of left steps taken by $W_1$ should equal the number of steps taken by $W_2$
The number of sequences of lefts and rights possible for each walker is $2^N$.
Now, in order for $W_1$ and $W_2$ to meet, they must have taken $0\ or\ 1\ or\ 2\ or\ 3\ or\ \ldots \ N $ left steps, which can be done in: $$ \sum_{i=0}^{N} \frac{\binom{N}{i}\binom{N}{i}}{2^N2^N} = \frac{\binom{2N}{N}}{4^N} $$