Two disjoint unit discs $D_1$ and $D_2$. Inside them there are random poisson points with intensity $\lambda$. For a given real $r>0$, what is the probability that there exist a poisson point $x_1\in D_1$ and a poisson point $x_2\in D_2$ such that $\x_1x_2\_2 \le r$? Couldn't find any existing results on this.
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If $M$ is the $4$dimensional Lebesgue measure of $\{(x,y) \in D_1 \times D_2: \x  y\_2 \le r\}$, then the number of such pairs of points is a Poisson random variable with parameter $M \lambda^2$, so the probability that there is at least one is $1  \exp(M \lambda^2)$. 

