
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.
from __future__ import division
from random import random
from math import fsum
 
n, p, t = 100, 0.5, 500
 
def newv(n, p): 
    return [int(random() < p) for i in range(n)]
 
def runs(v): 
    return sum((a & ~b) for a, b in zip(v, v[1:] + [0]))
 
def mean_run_density(n, p):
    return runs(newv(n, p)) / n
 
for p10 in range(1, 10, 2):
    p = p10 / 10
    limit = p * (1 - p)
    print('')
    for n2 in range(10, 16, 2):
        n = 2**n2
        sim = fsum(mean_run_density(n, p) for i in range(t)) / t
        print('t=%3i p=%4.2f n=%5i p(1-p)=%5.3f sim=%5.3f delta=%3.1f%%'
              % (t, p, n, limit, sim, abs(sim - limit) / limit * 100 if limit else sim * 100))