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

Crossing paths in 2D Random Walks