
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.
package main
 
import (
    "fmt"
    "math/rand"
)
 
var (
    pList = []float64{.1, .3, .5, .7, .9}
    nList = []int{1e2, 1e3, 1e4, 1e5}
    t     = 100
)
 
func main() {
    for _, p := range pList {
        theory := p * (1 - p)
        fmt.Printf("\np: %.4f  theory: %.4f  t: %d\n", p, theory, t)
        fmt.Println("        n          sim     sim-theory")
        for _, n := range nList {
            sum := 0
            for i := 0; i < t; i++ {
                run := false
                for j := 0; j < n; j++ {
                    one := rand.Float64() < p
                    if one && !run {
                        sum++
                    }
                    run = one
                }
            }
            K := float64(sum) / float64(t) / float64(n)
            fmt.Printf("%9d %15.4f %9.6f\n", n, K, K-theory)
        }
    }
}