HardcorePointProcess

HardcorePointProcess[μ,rh,d]

represents a hardcore point process with constant intensity μ and hardcore radius rh in .

Details

Examples

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Basic Examples  (2)

Sample from a hard-core point process in :

Visualize the points in the sample:

Sample from a hard-core point process defined on the surface of the Earth:

Visualize the points:

Scope  (3)

Generate three realizations from a hard-core point process in :

Estimate the parameters:

Generate three realizations from a hard-core point process on the surface of the Earth:

Visualize the point configurations:

Estimate the parameters:

Generate samples with increasing hard-core radius:

Plot samples with the repulsion disks:

Check that the hard-core constraint is obeyed:

Options  (4)

Method  (4)

Simulate using the Markov chain Monte Carlo method:

Specify the number of recursive calls to the sampler:

Specify the length of run:

Provide an initial state for the simulation:

The initial point must have nonzero density to ensure that the result is a valid configuration:

Check if the minimal distance between the points is smaller than the hardcore radius:

Visualize the birth and death process at different stages:

Use coupling from the past for exact sampling:

Properties & Relations  (3)

For the large intensity μ, the samples saturate:

The number of points saturates at a density that is significantly lower than the theoretical maximum packing:

Based on optimal packing :

The packing density:

For 3D:

Based on optimal packing :

The packing density:

Compute the average number of points in a unit disk for a hard-core point process:

Possible Issues  (1)

By default, the simulation will run until the number of points converges to a steady state, or until the default number of iterations is reached:

Raise the number of recursive calls to the sampler:

Increase the length of run:

Wolfram Research (2020), HardcorePointProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/HardcorePointProcess.html.

Text

Wolfram Research (2020), HardcorePointProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/HardcorePointProcess.html.

BibTeX

@misc{reference.wolfram_2021_hardcorepointprocess, author="Wolfram Research", title="{HardcorePointProcess}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/HardcorePointProcess.html}", note=[Accessed: 22-September-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_hardcorepointprocess, organization={Wolfram Research}, title={HardcorePointProcess}, year={2020}, url={https://reference.wolfram.com/language/ref/HardcorePointProcess.html}, note=[Accessed: 22-September-2021 ]}

CMS

Wolfram Language. 2020. "HardcorePointProcess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/HardcorePointProcess.html.

APA

Wolfram Language. (2020). HardcorePointProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HardcorePointProcess.html