
Percolation Simulation

This is a simulation of aspects of mathematical percolation theory.

For other percolation simulations, see Category:Percolation Simulations, or:
1D finite grid simulation
Mean run density 
2D finite grid simulations
Site percolation | Bond percolation | Mean cluster density

Let 



c


{\displaystyle c}

 be a 2D boolean square matrix of 



n
×
n


{\displaystyle n\times n}

 values of either 1 or 0 where the
probability of any value being 1 is 



p


{\displaystyle p}

, (and of 0 is therefore 



1
−
p


{\displaystyle 1-p}

).
We define a cluster of 1's as being a group of 1's connected vertically or
horizontally (i.e., using the Von Neumann neighborhood rule) and bounded by either 



0


{\displaystyle 0}

 or by the limits of the matrix.
Let the number of such clusters in such a randomly constructed matrix be 




C

n




{\displaystyle C_{n}}

.

Percolation theory states that 



K
(
p
)


{\displaystyle K(p)}

 (the mean cluster density) will satisfy 



K
(
p
)
=

C

n



/


n

2




{\displaystyle K(p)=C_{n}/n^{2}}

 as 



n


{\displaystyle n}

 tends to infinity. For 



p
=
0.5


{\displaystyle p=0.5}

, 



K
(
p
)


{\displaystyle K(p)}

 is found numerically to approximate 



0.065770


{\displaystyle 0.065770}

...

Task
Show the effect of varying 



n


{\displaystyle n}

 on the accuracy of simulated 



K
(
p
)


{\displaystyle K(p)}

 for 



p
=
0.5


{\displaystyle p=0.5}

 and
for values of 



n


{\displaystyle n}

 up to at least 



1000


{\displaystyle 1000}

.
Any calculation of 




C

n




{\displaystyle C_{n}}

 for finite 



n


{\displaystyle n}

 is subject to randomness, so an approximation should be
computed as the average of 



t


{\displaystyle t}

 runs, where 



t


{\displaystyle t}

 ≥ 



5


{\displaystyle 5}

.

For extra credit, graphically show clusters in a 



15
×
15


{\displaystyle 15\times 15}

, 



p
=
0.5


{\displaystyle p=0.5}

 grid.

Show your output here.

See also
 s-Cluster on Wolfram mathworld.
USING: combinators formatting generalizations kernel math
math.matrices random sequences ;
IN: rosetta-code.mean-cluster-density
 
CONSTANT: p 0.5
CONSTANT: iterations 5
 
: rand-bit-matrix ( n probability -- matrix )
    dupd [ random-unit > 1 0 ? ] curry make-matrix ;
 
: flood-fill ( x y matrix -- )
    3dup ?nth ?nth 1 = [
        [ [ -1 ] 3dip nth set-nth ] [
            {
                [ [ 1 + ] 2dip ]
                [ [ 1 - ] 2dip ]
                [ [ 1 + ] dip ]
                [ [ 1 - ] dip ]
            } [ flood-fill ] map-compose 3cleave
        ] 3bi
    ] [ 3drop ] if ;
 
: count-clusters ( matrix -- Cn )
    0 swap dup dim matrix-coordinates flip concat [
        first2 rot 3dup ?nth ?nth 1 = [ flood-fill 1 + ]
        [ 3drop ] if
    ] with each ;
 
: mean-cluster-density ( matrix -- mcd )
    [ count-clusters ] [ dim first sq / ] bi ;
 
: simulate ( n -- avg-mcd )
    iterations swap [ p rand-bit-matrix mean-cluster-density ]
    curry replicate sum iterations / ;
 
: main ( -- )
    { 4 64 256 1024 4096 } [
        [ iterations p ] dip dup simulate
        "iterations = %d p = %.1f n = %4d sim = %.5f\n" printf
    ] each ;
 
MAIN: main