
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.
Function run_test(p As Double, longitud As Integer, runs As Integer) As Double
    Dim As Integer r, l, cont = 0
    Dim As Integer v, pv
 
    For r = 1 To runs
        pv = 0
        For l = 1 To longitud
            v = Rnd < p
            cont += Iif(pv < v, 1, 0)
            pv = v
        Next l
    Next r
    Return (cont/runs/longitud)
End Function
 
Print "Running 1000 tests each:"
Print " p        n  K       p(1-p)       delta"
Print String(46,"-")
 
Dim As Double K, p, p1p
Dim As Integer n, ip
 
For ip = 1 To 10 Step 2
    p = ip / 10
    p1p = p * (1-p)
    n = 100
    While n <= 100000
        K = run_test(p, n, 1000)
        Print Using !"#.#  ######  #.####  #.####  +##.#### (##.## \b%)"; _
        p; n; K; p1p; K-p1p; (K-p1p)/p1p*100
        n *= 10
    Wend
    Print
Next ip
Sleep