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HardcorePointProcess
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HardcorePointProcess

[Experimental]

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represents a hard-core point process with constant intensity μ and hard-core radius rh in .

Details

Examples

open allclose all

Basic Examples  (2)Summary of the most common use cases

Sample from a hard-core point process in :

In[1]:=1
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https://wolfram.com/xid/0g38do4fqet8yf872-f9h16b
In[2]:=2
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https://wolfram.com/xid/0g38do4fqet8yf872-cr2aru
In[3]:=3
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https://wolfram.com/xid/0g38do4fqet8yf872-tfb0w
Out[3]=3

Visualize the points in the sample:

In[4]:=4
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https://wolfram.com/xid/0g38do4fqet8yf872-1pasjf
Out[4]=4

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

In[1]:=1
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https://wolfram.com/xid/0g38do4fqet8yf872-dm5er3
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0g38do4fqet8yf872-h6txtp
Out[2]=2

Visualize the points:

In[3]:=3
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https://wolfram.com/xid/0g38do4fqet8yf872-c96vjb
Out[3]=3

Scope  (3)Survey of the scope of standard use cases

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

In[1]:=1
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https://wolfram.com/xid/0g38do4fqet8yf872-pjmeyf
In[2]:=2
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https://wolfram.com/xid/0g38do4fqet8yf872-fat85p
In[3]:=3
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https://wolfram.com/xid/0g38do4fqet8yf872-y8f38b
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0g38do4fqet8yf872-kv2ge2
Out[4]=4

Estimate the parameters:

In[6]:=6
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https://wolfram.com/xid/0g38do4fqet8yf872-casa32
Out[6]=6

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

In[1]:=1
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https://wolfram.com/xid/0g38do4fqet8yf872-cw3l9q
In[2]:=2
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https://wolfram.com/xid/0g38do4fqet8yf872-duwsm2
In[3]:=3
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https://wolfram.com/xid/0g38do4fqet8yf872-o1wesh
Out[3]=3

Visualize the point configurations:

In[4]:=4
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https://wolfram.com/xid/0g38do4fqet8yf872-t6zgr
Out[4]=4

Estimate the parameters:

In[5]:=5
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https://wolfram.com/xid/0g38do4fqet8yf872-b50yar
Out[5]=5

Generate samples with increasing hard-core radius:

In[1]:=1
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https://wolfram.com/xid/0g38do4fqet8yf872-qpbanu
In[2]:=2
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https://wolfram.com/xid/0g38do4fqet8yf872-qrmox4
In[3]:=3
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https://wolfram.com/xid/0g38do4fqet8yf872-5qi0v
In[4]:=4
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https://wolfram.com/xid/0g38do4fqet8yf872-cuvxs8
Out[4]=4

Plot samples with the repulsion disks:

In[5]:=5
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https://wolfram.com/xid/0g38do4fqet8yf872-xgebf9
In[6]:=6
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https://wolfram.com/xid/0g38do4fqet8yf872-bq5vc4
Out[6]=6

Check that the hard-core constraint is obeyed:

In[7]:=7
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https://wolfram.com/xid/0g38do4fqet8yf872-g5f79f
In[8]:=8
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https://wolfram.com/xid/0g38do4fqet8yf872-bih4qk
Out[8]=8
In[9]:=9
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https://wolfram.com/xid/0g38do4fqet8yf872-n88e8
Out[9]=9

Options  (4)Common values & functionality for each option

Method  (4)

Simulate using the Markov chain Monte Carlo method:

In[15]:=15
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https://wolfram.com/xid/0g38do4fqet8yf872-mr431h
In[17]:=17
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https://wolfram.com/xid/0g38do4fqet8yf872-q096j
Out[17]=17

Specify the number of recursive calls to the sampler:

In[18]:=18
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https://wolfram.com/xid/0g38do4fqet8yf872-b28ma3
Out[18]=18

Specify the length of run:

In[19]:=19
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https://wolfram.com/xid/0g38do4fqet8yf872-hxjlyn
Out[19]=19

Provide an initial state for the simulation:

In[1]:=1
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https://wolfram.com/xid/0g38do4fqet8yf872-gx4n0o
In[2]:=2
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https://wolfram.com/xid/0g38do4fqet8yf872-f40juq
Out[2]=2

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

In[3]:=3
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https://wolfram.com/xid/0g38do4fqet8yf872-ggxwan
In[4]:=4
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https://wolfram.com/xid/0g38do4fqet8yf872-baijzk
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/0g38do4fqet8yf872-cdwlwh
Out[5]=5

Check if the minimal distance between the points is smaller than the hard-core radius:

In[6]:=6
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https://wolfram.com/xid/0g38do4fqet8yf872-vir4r
Out[6]=6
In[7]:=7
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https://wolfram.com/xid/0g38do4fqet8yf872-fqjy8c
Out[7]=7

Visualize the birth and death process at different stages:

In[1]:=1
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https://wolfram.com/xid/0g38do4fqet8yf872-iizyek
In[2]:=2
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https://wolfram.com/xid/0g38do4fqet8yf872-o4dprz
In[3]:=3
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https://wolfram.com/xid/0g38do4fqet8yf872-6e66ov
Out[3]=3

Use coupling from the past for exact sampling:

In[1]:=1
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https://wolfram.com/xid/0g38do4fqet8yf872-o474b1
In[2]:=2
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https://wolfram.com/xid/0g38do4fqet8yf872-ea30a8
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0g38do4fqet8yf872-kpeujq
Out[3]=3

Properties & Relations  (3)Properties of the function, and connections to other functions

For the large intensity μ, the samples saturate:

In[1]:=1
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https://wolfram.com/xid/0g38do4fqet8yf872-hux991
In[2]:=2
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https://wolfram.com/xid/0g38do4fqet8yf872-cmj10y
In[3]:=3
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https://wolfram.com/xid/0g38do4fqet8yf872-gzg6od
In[4]:=4
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https://wolfram.com/xid/0g38do4fqet8yf872-yyiq3s
In[5]:=5
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https://wolfram.com/xid/0g38do4fqet8yf872-kdvbpq
Out[5]=5

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

In[1]:=1
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https://wolfram.com/xid/0g38do4fqet8yf872-2feui
In[2]:=2
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https://wolfram.com/xid/0g38do4fqet8yf872-gjjp9n
In[3]:=3
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https://wolfram.com/xid/0g38do4fqet8yf872-8xvykp
Out[3]=3

Based on optimal packing :

In[4]:=4
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https://wolfram.com/xid/0g38do4fqet8yf872-cc51n0
Out[4]=4

The packing density:

In[5]:=5
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https://wolfram.com/xid/0g38do4fqet8yf872-pahnu
Out[5]=5

For 3D:

In[6]:=6
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https://wolfram.com/xid/0g38do4fqet8yf872-ch273w
In[7]:=7
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https://wolfram.com/xid/0g38do4fqet8yf872-cx0kdm
In[8]:=8
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https://wolfram.com/xid/0g38do4fqet8yf872-dqrgjn
Out[8]=8

Based on optimal packing :

In[9]:=9
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https://wolfram.com/xid/0g38do4fqet8yf872-hla1f
Out[9]=9

The packing density:

In[10]:=10
✖
https://wolfram.com/xid/0g38do4fqet8yf872-nop6t
Out[10]=10

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

In[1]:=1
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https://wolfram.com/xid/0g38do4fqet8yf872-ppr43z
In[2]:=2
✖
https://wolfram.com/xid/0g38do4fqet8yf872-mai9ud
Out[2]=2

Possible Issues  (1)Common pitfalls and unexpected behavior

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:

In[1]:=1
✖
https://wolfram.com/xid/0g38do4fqet8yf872-ja4d7q
In[3]:=3
✖
https://wolfram.com/xid/0g38do4fqet8yf872-bnb3wu
Out[3]=3

Raise the number of recursive calls to the sampler:

In[4]:=4
✖
https://wolfram.com/xid/0g38do4fqet8yf872-m520k
Out[4]=4

Increase the length of run:

In[5]:=5
✖
https://wolfram.com/xid/0g38do4fqet8yf872-hcyskp
Out[5]=5
Wolfram Research (2020), HardcorePointProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/HardcorePointProcess.html.
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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.

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

CMS

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

✖
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

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

BibTeX

@misc{reference.wolfram_2025_hardcorepointprocess, author="Wolfram Research", title="{HardcorePointProcess}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/HardcorePointProcess.html}", note=[Accessed: 09-July-2025 ]}

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@misc{reference.wolfram_2025_hardcorepointprocess, author="Wolfram Research", title="{HardcorePointProcess}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/HardcorePointProcess.html}", note=[Accessed: 09-July-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_hardcorepointprocess, organization={Wolfram Research}, title={HardcorePointProcess}, year={2020}, url={https://reference.wolfram.com/language/ref/HardcorePointProcess.html}, note=[Accessed: 09-July-2025 ]}

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@online{reference.wolfram_2025_hardcorepointprocess, organization={Wolfram Research}, title={HardcorePointProcess}, year={2020}, url={https://reference.wolfram.com/language/ref/HardcorePointProcess.html}, note=[Accessed: 09-July-2025 ]}