
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 



v


{\displaystyle v}

 be a vector of 



n


{\displaystyle n}

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



p


{\displaystyle p}

; the probability of a value being 0 is therefore 



1
−
p


{\displaystyle 1-p}

.
Define a run of 1s as being a group of consecutive 1s in the vector bounded
either by the limits of the vector or by a 0. Let the number of such runs in a given
vector of length 



n


{\displaystyle n}

 be 




R

n




{\displaystyle R_{n}}

.

For example, the following vector has 




R

10


=
3


{\displaystyle R_{10}=3}



[1 1 0 0 0 1 0 1 1 1]
 ^^^       ^   ^^^^^

Percolation theory states that





K
(
p
)
=

lim

n
→
∞



R

n



/

n
=
p
(
1
−
p
)


{\displaystyle K(p)=\lim _{n\to \infty }R_{n}/n=p(1-p)}


Task
Any calculation of 




R

n



/

n


{\displaystyle R_{n}/n}

 for finite 



n


{\displaystyle n}

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



t


{\displaystyle t}

 runs, where 



t
≥
100


{\displaystyle t\geq 100}

.

For values of 



p


{\displaystyle p}

 of 0.1, 0.3, 0.5, 0.7, and 0.9, show the effect of varying 



n


{\displaystyle n}


on the accuracy of simulated 



K
(
p
)


{\displaystyle K(p)}

.

Show your output here.

See also
 s-Run on Wolfram mathworld.
import "random" for Random
import "/fmt" for Fmt
 
var rand = Random.new()
var RAND_MAX = 32767
 
// just generate 0s and 1s without storing them
var runTest = Fn.new { |p, len, runs|
    var cnt = 0
    var thresh = (p * RAND_MAX).truncate
    for (r in 0...runs) {
        var x = 0
        var i = len
        while (i > 0) {
            i = i - 1
            var y = (rand.int(RAND_MAX + 1) < thresh) ? 1 : 0
            if (x < y) cnt = cnt + 1
            x = y
        }
    }
    return cnt / runs / len
}
 
System.print("Running 1000 tests each:")
System.print(" p\t   n\tK\tp(1-p)\t     diff")
System.print("------------------------------------------------")
var fmt = "$.1f\t$6d\t$.4f\t$.4f\t$+.4f ($+.2f\%)"
for (ip in [1, 3, 5, 7, 9]) {
    var p = ip / 10
    var p1p = p * (1 - p)
    var n = 100
    while (n <= 1e5) {
        var k = runTest.call(p, n, 1000)
        Fmt.lprint(fmt, [p, n, k, p1p, k - p1p, (k - p1p) /p1p * 100])
        n = n * 10
    }
    System.print()
}