When it comes to Poisson Point Process, it is a straightforward task to formulate the contact distance distribution of a point located at the origin. However, I am wondering if there is a similar formula describing the contact distance in a Hard-Core Matérn point process (Type-II).
See Martin Haenggi "Stochastic Geometry for Wireless Networks", Cambridge University Press, 2013.
Dear Akram,
The contact distribution for type 2 hardcore process cannot be expressed in closed form. However in a certain regime the type 2 hardcore process can be easily approximated as equi-dense PPP. Thus the distance distribution from HPPP can be used. Please see following papers:
http://arxiv.org/pdf/1103.1598.pdf
http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6574907
The approximation between PPP and Matérn is addressed interms of the interference. However is there more closely related work specifically addressing the contact distance in Matérn type-II ? is there an empirical formula for the same ?
Hi Akram, Please see Eq. (2) in the second link. This can be treated as a contact distribution as it specifies the probability that two points within r distance of each other are retained, which is same as saying that two points are separated by distance r in Matern typeII process.
Thanks for your answer. The contact distance PDF and the retention probability should not be the same. Since the former is a PDF and the latter is a "probability". Another reason, is that the referred equation is constant when (r > 2 * delta), which is not valid for a PDF. I think there is no closed form yet discovered for the contact distance in Matern Hard Core Process. But is there an empirical formula ?
Another thing, I should be clearer in the question, i am referring by the contact distance to the distance between a point (A) and the nearest point in the the MHCP, where (A) is a an external point that is dropped randomly in the region. N.B. In the case of PPP the contact disatance and the nearets neighbor are the same.
I meant you can integrate the k(r) to have CDF. However, from your question the issue is not about contact distribution of the Matern hard-core process exactly. What you mean is that what is the distance of the randomly dropped point to the nearest neighbour. Given that the point is not the part of hard-core process it can be approximated easily by the contact distribution of PPP with the density Lambda*retention probability.
We have solved the contact distance problem in Matérn Hard-Core Point Process. please see the below document:
"Nearest Neighbour Distance Distribution in Hard-Core Point Processes"
http://arxiv.org/pdf/1606.03695v1.pdf
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