I am looking for the probability that two random walkers initially at different sites, meet at step t if they are moving on a 2-dimensional torus(Square Lattice)
Any help would be appreciated.
I am looking for the probability that two random walkers initially at different sites, meet at step t if they are moving on a 2-dimensional torus(Square Lattice)
Any help would be appreciated.
One idea would be ot fix one random walker and look a the law of the difference, i.e., instead of looking at the distance between $X_1$ and $X_2$, you look at the distance between $X_1-X_2$ and the origin. But in here, since the two walkers have the same law and are (I guess) independent, you can write: $$\begin{align} P_{(a,b)}(X_1(t)=X_2(t))&=\sum_ {x\in T}P_a(X_1(t)=x)\times P_b(X_2(t)=x) \\ &=\sum_ {x\in T}P_a(X_1(t)=x)\times P_x(X_1(t)=b) \\ &=P_a(X_1(2t)=b) \\ \end{align}$$ where $P_{(a,b)}$ means that $X_1(0)=a$ and $X_2(0)=b$, and $T$ is the set of vertices of the torus.