
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.
with javascript_semantics
function run_test(atom p, integer len, runs)
    integer count = 0
    for r=1 to runs do
        bool v, pv = false
        for l=1 to len do
            v = rnd()<p
            count += pv<v
            pv = v
        end for
    end for
    return count/runs/len
end function
 
procedure main()
    printf(1,"Running 1000 tests each:\n")
    printf(1," p        n  K       p(1-p)       delta\n")
    printf(1,"--------------------------------------------\n")
    for ip=1 to 10 by 2 do
        atom p = ip/10,
             p1p = p*(1-p)
        integer n = 100
        while n<=100000 do
            atom K = run_test(p, n, 1000)
            printf(1,"%.1f  %6d  %6.4f  %6.4f  %+7.4f (%+5.2f%%)\n",
                   {p, n, K, p1p, K-p1p, (K-p1p)/p1p*100})
            n *= 10
        end while
        printf(1,"\n")
    end for
end procedure
main()
