
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.
UInt32 seed = 0
F nonrandom()
   :seed = 1664525 * :seed + 1013904223
   R Int(:seed >> 16) / Float(FF'FF)
 
V (p, t) = (0.5, 500)
 
F newv(n, p)
   R (0 .< n).map(i -> Int(nonrandom() < @p))
 
F runs(v)
   R sum(zip(v, v[1..] [+] [0]).map((a, b) -> (a [&] (-)b)))
 
F mean_run_density(n, p)
   R runs(newv(n, p)) / Float(n)
 
L(p10) (1.<10).step(2)
   p = p10 / 10
   V limit = p * (1 - p)
   print(‘’)
   L(n2) (10.<16).step(2)
      V n = 2 ^ n2
      V sim = sum((0 .< t).map(i -> mean_run_density(@n, :p))) / t
      print(‘t=#3 p=#.2 n=#5 p(1-p)=#.3 sim=#.3 delta=#.1%’.format(
             t,   p,    n,       limit, sim, I limit {abs(sim - limit) / limit * 100} E sim * 100))