
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 Control.Monad.Random
import Control.Applicative
import Text.Printf
import Control.Monad
import Data.Bits
 
data OneRun = OutRun | InRun deriving (Eq, Show)
 
randomList :: Int -> Double -> Rand StdGen [Int]
randomList n p = take n . map f <$> getRandomRs (0,1)
  where f n = if (n > p) then 0 else 1
 
countRuns xs = fromIntegral . sum $ 
               zipWith (\x y -> x .&. xor y 1) xs (tail xs ++ [0])
 
calcK :: Int -> Double -> Rand StdGen Double
calcK n p = (/ fromIntegral n) . countRuns <$> randomList n p
 
printKs :: StdGen -> Double -> IO ()
printKs g p = do
  printf "p= %.1f, K(p)= %.3f\n" p (p * (1 - p))
  forM_ [1..5] $ \n -> do
    let est = evalRand (calcK (10^n) p) g
    printf "n=%7d, estimated K(p)= %5.3f\n" (10^n::Int) est
 
main = do
  x <- newStdGen
  forM_ [0.1,0.3,0.5,0.7,0.9] $ printKs x
 