Crossing paths in 2D Random Walks

We investigate crossing path probabilities for two agents that move randomly in a bounded region of the plane or on a sphere (denoted $R$). At each discrete time-step the agents move, independently, fixed distances $d_1$ and $d_2$ at angles that are uniformly distributed in $(0,2\pi)$. If $R$ is large enough and the initial positions of the agents are uniformly distributed in $R$, then the probability of paths crossing at the first time-step is close to $2d_1d_2/(\pi A[R])$, where $A[R]$ is the area of $R$. Simulations suggest that the long-run rate at which paths cross is also close to $2d_1d_2/(\pi A[R])$ (despite marked departures from uniformity and independence conditions needed for such a conclusion).