
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.
proc randomString {length probability} {
    for {set s ""} {[string length $s] < $length} {} {
	append s [expr {rand() < $probability}]
    }
    return $s
}
 
# By default, [regexp -all] gives the number of times that the RE matches
proc runs {str} {
    regexp -all {1+} $str
}
 
# Compute the mean run density
proc mrd {t p n} {
    for {set i 0;set total 0.0} {$i < $t} {incr i} {
	set run [randomString $n $p]
	set total [expr {$total + double([runs $run])/$n}]
    }
    return [expr {$total / $t}]
}
 
# Parameter sweep with nested [foreach]
set runs 500
foreach p {0.10 0.30 0.50 0.70 0.90} {
    foreach n {1024 4096 16384} {
	set theory [expr {$p * (1 - $p)}]
	set sim [mrd $runs $p $n]
	set diffpc [expr {abs($theory-$sim)*100/$theory}]
	puts [format "t=%d, p=%.2f, n=%5d, p(1-p)=%.3f, sim=%.3f, delta=%.2f%%" \
		      $runs $p $n $theory $sim $diffpc]
    }
    puts ""
}