Simulation of the Second Matern Hard Core point process Using the second matern point process thinning method, I have obtained a hard core point process from a regular poisson process in $\mathbb{R}^2$. According to the standard text (Stochastic Geometry and its Applications), the new intensity of the thinned point process is 
\begin{equation}
\lambda = \frac{1 - exp(-\lambda_{PP}\pi r^2)}{\pi r^2}
\end{equation}
where, $\lambda_{PP}$ represents the intensity of the original Poisson process and $r$ is the hard core distance between the points. I understand that $\lambda$ signifies the $\textbf{expected number of points in a region of unit area}$. I want to compute the probability of having at-least one point in a region of area $A_r$. Treating this like a poisson process, I get 
\begin{equation}
\phi = 1 - exp(-\lambda A_r)
\end{equation}
I have been unable to verify this probability calculation using simple simulations. The probability estimate obtained after many iterations is significantly higher than the one predicted by the above formula. Any idea why this might be happening? 
 A: You can't treat the Matérn II point process like a Poisson point process, if that's what you are doing. That expression (for the complement of the void probability  or avoidance function) is only valid for the Poisson point process. 
More generally, it's difficult to answer the question without seeing the code. I have found that the main challenge behind sampling the Matérn hard-core point processes is that it's possible for points inside the simulation window to be thinned due to their closeness to points that are located outside the simulation window. In other words, points outside the simulation window can cause points inside the window to be thinned. Such effects due to points from outside the simulation window can be considered as "edge effects". And people often forget to include the edge effects, although for large simulation windows, these effects will become increasingly negligible.
I wrote in MATLAB and Python code for simulating/sampling Matern Type I and Type II point processes. This code incorporates the edge effects by simulating the underlying Poisson point process on an extended window. The empirical results for the intensity of the hard-core point processes agree with the analytic expressions. 
The code is available here:
https://github.com/hpaulkeeler/posts/tree/master/MaternHardcoreRectangle
Further details on simulating Matérn hard-core point processes are detailed in this post:
https://hpaulkeeler.com/simulating-matern-hard-core-point-processes/
For more information, I suggest looking at the recommended citations at the end of the post.
