
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.
const X=-1;	// the sentinal that marks an untouched cell
var C,N,NN,P;
fcn createC(n,p){
   N,P=n,p; NN=N*N;
   C=NN.pump(List.createLong(NN),0);  // vector of ints
   foreach n in (NN){ C[n]=X*(Float.random(1)<=P) } // X is the sentinal
}
fcn showCluster{
   alpha:="-ABCDEFGHIJKLMNOPQRSTUVWXYZ" "abcdefghijklmnopqrstuvwxyz";
   foreach n in ([0..NN,N]){ C[n,N].pump(String,alpha.get).println() }
}
fcn countClusters{
   clusters:=0;
   foreach n in (NN){ 
      if(X!=C[n]) continue;
      fcn(n,v){
	 if((0<=n<NN) and C[n]==X){
	    C[n]=v;
	    self.fcn(n-N,v); self.fcn(n-1,v); self.fcn(n+1,v); self.fcn(n+N,v);
	 }
      }(n,clusters+=1);
   }
   clusters
}
fcn tests(N,n,p){
   k:=0.0;
   foreach z in (n){ createC(N,p); k+=countClusters().toFloat()/NN; }
   k/n
}