Let us consider a Poisson Point Process (PPP) $\Phi$ of density $\lambda$.
The region is a unit circle (radius=1).
If there is a point $o\notin\Phi$ (not belonging to the poison process) in the centre, how to find the minimum distance from the point $o$ to the closest point $z$ belonging to the PPP, i.e., $z\in\Phi$ for different $\lambda$?
Note: The distribution of the distance between the point $o$ and $z$, i.e.,$r=\|o-z\|$ is given by $$2\pi\lambda r\exp(-\pi\lambda r^2)$$
For example, I want $r_{\min}$ in terms of a numerical value or an expression. Lets say $r_{\min}=0.12$. Expectation will also be fine.
Best Answer
First I'll do this on the whole plane. Let $R$ be the distance from the origin to the nearest point in the process. Then \begin{align} & \Pr(R \ge r) \\[6pt] = {} & \Pr(\text{the number of points in the process at distance $<r$ from the origin} = 0) \\[6pt] = {} & \frac{(\lambda\cdot\text{area})^0 e^{-\lambda\cdot\text{area}}}{0!} = e^{-\lambda\pi r^2}. \tag 1 \end{align} From this you can find the c.d.f. and the p.d.f.
Now let's assume the support is a disk. Here I'm tempted to say $R=\infty$ if the number of points or the process that are within the disk are $0$. Alternatively, one could plausibly say it is the radius of the disk. In that case $(1)$ would hold for $r\le\text{the radius}$ and you've have to work piecewise; you'd have a mixture of discrete and continuous distributions.