
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.
import std.stdio, std.range, std.algorithm, std.random, std.math;
 
enum n = 100, p = 0.5, t = 500;
 
double meanRunDensity(in size_t n, in double prob) {
    return n.iota.map!(_ => uniform01 < prob)
           .array.uniq.sum / double(n);
}
 
void main() {
    foreach (immutable p; iota(0.1, 1.0, 0.2)) {
        immutable limit = p * (1 - p);
        writeln;
        foreach (immutable n2; iota(10, 16, 2)) {
            immutable n = 2 ^^ n2;
            immutable sim = t.iota.map!(_ => meanRunDensity(n, p))
                            .sum / t;
            writefln("t=%3d, p=%4.2f, n=%5d, p(1-p)=%5.5f, " ~
                     "sim=%5.5f, delta=%3.1f%%", t, p, n, limit, sim,
                     limit ? abs(sim - limit) / limit * 100 : sim*100);
        }
    }
}