Percolation/Site percolation

Given an $M\times N$ rectangular array of cells numbered $\mathrm {cell} [0..M-1,0..N-1]$assume $M$ is horizontal and $N$ is downwards.

Percolation/Site percolation
You are encouraged to solve this task according to the task description, using any language you may know.

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

Assume that the probability of any cell being filled is a constant $p$ where

$0.0\leq p\leq 1.0$

The task

Simulate creating the array of cells with probability $p$ and then testing if there is a route through adjacent filled cells from any on row $0$ to any on row $N$, i.e. testing for site percolation.

Given $p$ repeat the percolation $t$ times to estimate the proportion of times that the fluid can percolate to the bottom for any given $p$.

Show how the probability of percolating through the random grid changes with $p$ going from $0.0$ to $1.0$ in $0.1$ increments and with the number of repetitions to estimate the fraction at any given $p$ as $t>=100$.

Use an $M=15,N=15$ grid of cells for all cases.

Optionally depict a percolation through a cell grid graphically.

Show all output on this page.

C

<lang c>#include <stdio.h>

1. include <stdlib.h>
2. include <string.h>

char *cell, *start, *end; int m, n;

void make_grid(int x, int y, double p) { int i, j, thresh = p * RAND_MAX;

m = x, n = y; end = start = realloc(start, (x+1) * (y+1) + 1);

memset(start, 0, m + 1);

cell = end = start + m + 1; for (i = 0; i < n; i++) { for (j = 0; j < m; j++) *end++ = rand() < thresh ? '+' : '.'; *end++ = '\n'; }

end[-1] = 0; end -= ++m; // end is the first cell of bottom row }

int ff(char *p) // flood fill { if (*p != '+') return 0;

*p = '#'; return p >= end || ff(p+m) || ff(p+1) || ff(p-1) || ff(p-m); }

int percolate(void) { int i; for (i = 0; i < m && !ff(cell + i); i++); return i < m; }

int main(void) { make_grid(15, 15, .5); percolate();

puts("15x15 grid:"); puts(cell);

puts("\nrunning 10,000 tests for each case:");

double p; int ip, i, cnt; for (ip = 0; ip <= 10; ip++) { p = ip / 10.; for (cnt = i = 0; i < 10000; i++) { make_grid(15, 15, p); cnt += percolate(); } printf("p=%.1f:  %.4f\n", p, cnt / 10000.); }

return 0; }</lang>

Output:

15x15 grid:
.#...##.#.#.#..
...+.###.####.#
...+..#.+...#.#
+..+..##..#####
+...+.#....##..
.+..+.##..##.+.
....+.#...##..+
..+.+.#####.++.
+++....#.###.++
.+.+.#.#.##....
..++.####...++.
+.+.+.##..+++..
+..+.+..+.....+
..........++..+
.+.+.++++.+...+

running 10,000 tests for each case:
p=0.0:  0.0000
p=0.1:  0.0000
p=0.2:  0.0000
p=0.3:  0.0000
p=0.4:  0.0032
p=0.5:  0.0902
p=0.6:  0.5771
p=0.7:  0.9587
p=0.8:  0.9996
p=0.9:  1.0000
p=1.0:  1.0000

Translation of: D

<lang c>#include <stdio.h>

1. include <stdlib.h>
2. include <time.h>
3. include <string.h>
4. include <stdbool.h>

1. define N_COLS 15
2. define N_ROWS 15

// Probability granularity 0.0, 0.1, ... 1.0

1. define N_STEPS 11

// Simulation tries

1. define N_TRIES 100

typedef unsigned char Cell; enum { EMPTY_CELL = ' ',

      FILLED_CELL  = '#',
      VISITED_CELL = '.' };

typedef Cell Grid[N_ROWS][N_COLS];

void initialize(Grid grid, const double probability) {

   for (size_t r = 0; r < N_ROWS; r++)
       for (size_t c = 0; c < N_COLS; c++) {
           const double rnd = rand() / (double)RAND_MAX;
           grid[r][c] = (rnd < probability) ? EMPTY_CELL : FILLED_CELL;
       }

}

void show(Grid grid) {

   char line[N_COLS + 3];
   memset(&line[0], '-', N_COLS + 2);
   line[0] = '+';
   line[N_COLS + 1] = '+';
   line[N_COLS + 2] = '\0';

   printf("%s\n", line);
   for (size_t r = 0; r < N_ROWS; r++) {
       putchar('|');
       for (size_t c = 0; c < N_COLS; c++)
           putchar(grid[r][c]);
       puts("|");
   }
   printf("%s\n", line);

}

bool walk(Grid grid, const size_t r, const size_t c) {

   const size_t bottom = N_ROWS - 1;
   grid[r][c] = VISITED_CELL;

   if (r < bottom && grid[r + 1][c] == EMPTY_CELL) { // Down.
       if (walk(grid, r + 1, c))
           return true;
   } else if (r == bottom)
       return true;

   if (c && grid[r][c - 1] == EMPTY_CELL) // Left.
       if (walk(grid, r, c - 1))
           return true;

   if (c < N_COLS - 1 && grid[r][c + 1] == EMPTY_CELL) // Right.
       if (walk(grid, r, c + 1))
           return true;

   if (r && grid[r - 1][c] == EMPTY_CELL) // Up.
       if (walk(grid, r - 1, c))
           return true;

   return false;

}

bool percolate(Grid grid) {

   const size_t startR = 0;
   for (size_t c = 0; c < N_COLS; c++)
       if (grid[startR][c] == EMPTY_CELL)
           if (walk(grid, startR, c))
               return true;
   return false;

}

typedef struct {

   double prob;
   size_t count;

} Counter;

int main() {

   const double probability_step = 1.0 / (N_STEPS - 1);
   Counter counters[N_STEPS];
   for (size_t i = 0; i < N_STEPS; i++)
       counters[i] = (Counter){ i * probability_step, 0 };

   bool sample_shown = false;
   static Grid grid;
   srand(time(NULL));

   for (size_t i = 0; i < N_STEPS; i++) {
       for (size_t t = 0; t < N_TRIES; t++) {
           initialize(grid, counters[i].prob);
           if (percolate(grid)) {
               counters[i].count++;
               if (!sample_shown) {
                   printf("Percolating sample (%dx%d,"
                          " probability =%5.2f):\n",
                          N_COLS, N_ROWS, counters[i].prob);
                   show(grid);
                   sample_shown = true;
               }
           }
       }
   }

   printf("\nFraction of %d tries that percolate through:\n", N_TRIES);
   for (size_t i = 0; i < N_STEPS; i++)
       printf("%1.1f %1.3f\n", counters[i].prob,
              counters[i].count / (double)N_TRIES);

   return 0;

} </lang>

Output:

Percolating sample (15x15, probability = 0.40):
+---------------+
|###.  #  # #  #|
|###.. #  ##### |
|   #. ######  #|
|###....  ######|
|######.  ### # |
| #####.######  |
|#......... ##  |
|...#...##.# ## |
|##.#...##.### #|
| ###..# #. #   |
|# #######. # ##|
|   # ##...#### |
| ##  # .#####  |
|#######.##  ###|
|# ##   .## # # |
+---------------+

Fraction of 100 tries that percolate through:
0.0 0.000
0.1 0.000
0.2 0.000
0.3 0.000
0.4 0.010
0.5 0.070
0.6 0.630
0.7 0.970
0.8 1.000
0.9 1.000
1.0 1.000

D

Translation of: Python

<lang d>import std.stdio, std.random, std.array, std.datetime;

enum size_t nCols = 15,

           nRows = 15,
           nSteps = 11,     // Probability granularity.
           nTries = 20_000; // Simulation tries.

enum Cell : char { empty = ' ', filled = '#', visited = '.' } alias Grid = Cell[nCols][nRows];

void initialize(ref Grid grid, in double probability, ref Xorshift rng) {

   foreach (ref row; grid)
       foreach (ref cell; row)
           cell = (rng.uniform01 < probability) ? Cell.empty : Cell.filled;

}

void show(in ref Grid grid) @safe {

   writefln("%(|%(%c%)|\n%)|", grid);

}

bool percolate(ref Grid grid) pure nothrow @safe @nogc {

   bool walk(in size_t r, in size_t c) nothrow @safe @nogc {
       enum bottom = nRows - 1;
       grid[r][c] = Cell.visited;

       if (r < bottom && grid[r + 1][c] == Cell.empty) { // Down.
           if (walk(r + 1, c))
               return true;
       } else if (r == bottom)
           return true;

       if (c && grid[r][c - 1] == Cell.empty) // Left.
           if (walk(r, c - 1))
               return true;

       if (c < nCols - 1 && grid[r][c + 1] == Cell.empty) // Right.
           if (walk(r, c + 1))
               return true;

       if (r && grid[r - 1][c] == Cell.empty) // Up.
           if (walk(r - 1, c))
               return true;

       return false;
   }

   enum startR = 0;
   foreach (immutable c; 0 .. nCols)
       if (grid[startR][c] == Cell.empty)
           if (walk(startR, c))
               return true;
   return false;

}

void main() {

   static struct Counter {
       double prob;
       size_t count;
   }

   StopWatch sw;
   sw.start;

   enum probabilityStep = 1.0 / (nSteps - 1);
   Counter[nSteps] counters;
   foreach (immutable i, ref co; counters)
       co.prob = i * probabilityStep;

   Grid grid;
   bool sampleShown = false;
   auto rng = Xorshift(unpredictableSeed);

   foreach (ref co; counters) {
       foreach (immutable _; 0 .. nTries) {
           grid.initialize(co.prob, rng);
           if (grid.percolate) {
               co.count++;
               if (!sampleShown) {
                   writefln("Percolating sample (%dx%d, probability =%5.2f):",
                            nCols, nRows, co.prob);
                   grid.show;
                   sampleShown = true;
               }
           }
       }
   }
   sw.stop;

   writefln("\nFraction of %d tries that percolate through:", nTries);
   foreach (const co; counters)
       writefln("%1.3f %1.3f", co.prob, co.count / double(nTries));

   writefln("\nSimulations and grid printing performed" ~
            " in %3.2f seconds.", sw.peek.msecs / 1000.0);

}</lang>

Output:

Percolating sample (15x15, probability = 0.40):
|#.###.##..#. # |
|#.###.# ###.  #|
|#.##..#####. ##|
|## ####  ...# #|
|# # #  ##.#..##|
|### # ## .#####|
|   ######.## ##|
|    ## #..###  |
|#### ##..##### |
|#   ###...  #  |
|### ## ##.   # |
|# ###  ##. ### |
|## ##### . ####|
|# ## #  #. ####|
|####### #.## ##|

Fraction of 20000 tries that percolate through:
0.000 0.000
0.100 0.000
0.200 0.000
0.300 0.000
0.400 0.004
0.500 0.090
0.600 0.565
0.700 0.958
0.800 1.000
0.900 1.000
1.000 1.000

Simulations and grid printing performed in 0.70 seconds.

Fortran

Please see sample compilation and program execution in comments at top of program. Thank you. This example demonstrates recursion and integer constants of a specific kind. <lang fortran> ! loosely translated from python. ! compilation: gfortran -Wall -std=f2008 thisfile.f08

!$ a=site && gfortran -o $a -g -O0 -Wall -std=f2008 $a.f08 && $a !100 trials per !Fill Fraction goal(%) simulated through paths(%) ! 0 0 ! 10 0 ! 20 0 ! 30 0 ! 40 0 ! 50 6 ! ! ! b b b b h j m m m ! b b b b b h h m m m m m ! b b b h h h m ! b h h h h h h h ! b b h h h h h h h h h ! b b b h h h h h h h h h h ! b b @ h h h h h h h ! @ @ h h h h h h h h ! @ @ @ @ h h h h ! @ @ @ @ h h h h h h ! @ @ @ h h h h h h h ! @ @ @ h h h h h h ! @ h h h h h h ! @ h h h h h h h ! @ @ h h h h h h h h h h ! 60 59 ! 70 97 ! 80 100 ! 90 100 ! 100 100

program percolation_site

 implicit none
 integer, parameter :: m=15,n=15,t=100
 !integer, parameter :: m=2,n=2,t=8
 integer(kind=1), dimension(m, n) :: grid
 real :: p
 integer :: i, ip, trial, successes
 logical :: success, unseen, q
 data unseen/.true./
 write(6,'(i3,a11)') t,' trials per'
 write(6,'(a21,a30)') 'Fill Fraction goal(%)','simulated through paths(%)'
 do ip=0, 10
    p = ip/10.0
    successes = 0
    do trial = 1, t
       call newgrid(grid, p)
       success = .false.
       do i=1, m
          q = walk(grid, i)    ! deliberately compute all paths
          success = success .or. q
       end do
       if ((ip == 6) .and. unseen) then
          call display(grid)
          unseen = .false.
       end if
       successes = successes + merge(1, 0, success)
    end do
    write(6,'(9x,i3,24x,i3)')ip*10,nint(100*real(successes)/real(t))
 end do

contains

 logical function walk(grid, start)
   integer(kind=1), dimension(m,n), intent(inout) :: grid
   integer, intent(in) :: start
   walk = rwalk(grid, 1, start, int(start+1,1))
 end function walk

 recursive function rwalk(grid, i, j, k) result(through)
   logical :: through
   integer(kind=1), dimension(m,n), intent(inout) :: grid
   integer, intent(in) :: i, j
   integer(kind=1), intent(in) :: k
   logical, dimension(4) :: q
   !out of bounds
   through = .false.
   if (i < 1) return
   if (m < i) return
   if (j < 1) return
   if (n < j) return
   !visited or non-pore
   if (1_1 /= grid(i, j)) return
   !update grid and recurse with neighbors.  deny 'shortcircuit' evaluation
   grid(i, j) = k
   q(1) = rwalk(grid,i+0,j+1,k)
   q(2) = rwalk(grid,i+0,j-1,k)
   q(3) = rwalk(grid,i+1,j+0,k)
   q(4) = rwalk(grid,i-1,j+0,k)
   !newly discovered outlet
   through = (i == m) .or. any(q)
 end function rwalk

 subroutine newgrid(grid, probability)
   implicit none
   real :: probability
   integer(kind=1), dimension(m,n), intent(out) :: grid
   real, dimension(m,n) :: harvest
   call random_number(harvest)
   grid = merge(1_1, 0_1, harvest < probability)
 end subroutine newgrid

 subroutine display(grid)
   integer(kind=1), dimension(m,n), intent(in) :: grid
   integer :: i, j, k, L
   character(len=n*2) :: lineout
   write(6,'(/)')
   lineout = ' '
   do i=1,m
      do j=1,n
         k = j+j
         L = grid(i,j)+1
         lineout(k:k) = ' @abcdefghijklmnopqrstuvwxyz'(L:L)
      end do
      write(6,*) lineout
   end do
 end subroutine display

end program percolation_site </lang>

Go

<lang go>package main

import ( "bytes" "fmt" "math/rand" "time" )

func main() { const ( m, n = 15, 15 t = 1e4 minp, maxp, Δp = 0, 1, 0.1 )

rand.Seed(2) // Fixed seed for repeatable example grid g := NewGrid(.5, m, n) g.Percolate() fmt.Println(g)

rand.Seed(time.Now().UnixNano()) // could pick a better seed for p := float64(minp); p < maxp; p += Δp { count := 0 for i := 0; i < t; i++ { g := NewGrid(p, m, n) if g.Percolate() { count++ } } fmt.Printf("p=%.2f, %.4f\n", p, float64(count)/t) } }

const ( full = '.' used = '#' empty = ' ' )

type grid struct { cell [][]byte // row first, i.e. [y][x] }

func NewGrid(p float64, xsize, ysize int) *grid { g := &grid{cell: make([][]byte, ysize)} for y := range g.cell { g.cell[y] = make([]byte, xsize) for x := range g.cell[y] { if rand.Float64() < p { g.cell[y][x] = full } else { g.cell[y][x] = empty } } } return g }

func (g *grid) String() string { var buf bytes.Buffer // Don't really need to call Grow but it helps avoid multiple // reallocations if the size is large. buf.Grow((len(g.cell) + 2) * (len(g.cell[0]) + 3))

buf.WriteByte('+') for _ = range g.cell[0] { buf.WriteByte('-') } buf.WriteString("+\n")

for y := range g.cell { buf.WriteByte('|') buf.Write(g.cell[y]) buf.WriteString("|\n") }

buf.WriteByte('+') ly := len(g.cell) - 1 for x := range g.cell[ly] { if g.cell[ly][x] == used { buf.WriteByte(used) } else { buf.WriteByte('-') } } buf.WriteByte('+') return buf.String() }

func (g *grid) Percolate() bool { for x := range g.cell[0] { if g.use(x, 0) { return true } } return false }

func (g *grid) use(x, y int) bool { if y < 0 || x < 0 || x >= len(g.cell[0]) || g.cell[y][x] != full { return false // Off the edges, empty, or used } g.cell[y][x] = used if y+1 == len(g.cell) { return true // We're on the bottom }

// Try down, right, left, up in that order. return g.use(x, y+1) || g.use(x+1, y) || g.use(x-1, y) || g.use(x, y-1) }</lang>

Output:

+---------------+
|####  ###.  .. |
| ##   # #   . .|
| ###   #### .. |
|### ##### #..  |
|   ### # ## .. |
|# ##     # . ..|
|### .    #.. . |
| ##      ##. ..|
| ## .. .. # .. |
| ##    . .#....|
|##   .. .##  . |
|# .  . . # .   |
| ..   . .#. .. |
|. . .... #  .. |
| . ..  . # .. .|
+---------#-----+
p=0.00, 0.0000
p=0.10, 0.0000
p=0.20, 0.0000
p=0.30, 0.0000
p=0.40, 0.0040
p=0.50, 0.0980
p=0.60, 0.5641
p=0.70, 0.9583
p=0.80, 0.9995
p=0.90, 1.0000
p=1.00, 1.0000

Haskell

<lang haskell> {-# LANGUAGE OverloadedStrings #-} import Control.Monad import Control.Monad.Random import Data.Array.Unboxed import Data.List import Formatting

type Field = UArray (Int, Int) Char

-- Start percolating some seepage through a field. -- Recurse to continue percolation with spreading seepage. percolateR :: [(Int, Int)] -> Field -> (Field, [(Int,Int)]) percolateR [] f = (f, []) percolateR seep f = percolateR

                      (concat $ fmap neighbors validSeep) 
                      (f // map (\p -> (p,'.')) validSeep) where
   neighbors p@(r,c) = [(r-1,c), (r+1,c), (r, c-1), (r, c+1)]
   ((rLo,cLo),(rHi,cHi)) = bounds f
   validSeep = filter (\p@(r,c) -> r >= rLo && 
                                   r <= rHi && 
                                   c >= cLo && 
                                   c <= cHi && 
                                   f!p == ' ') $ nub $ sort seep

-- Percolate a field; Return the percolated field. percolate :: Field -> Field percolate start =

   let ((_,_),(_,cHi)) = bounds start
       (final, _) = percolateR [(0,c) | c <- [0..cHi]] start
   in final

-- Generate a random field. randomField :: Int -> Int -> Double -> Rand StdGen Field randomField rows cols threshold = do

   rnd <- replicateM rows (replicateM cols $ getRandomR (0.0, 1.0))
   return $ array ((0,0), (rows-1, cols-1)) 
                   [((r,c), if rnd !! r !! c < threshold then ' ' 
                            else '#') 
                    | r <- [0..rows-1], c <- [0..cols-1] ] 

-- Assess whether or not percolation reached bottom of field. leaky :: Field -> Bool leaky f = '.' `elem` [f!(rHi,c) | c <- [cLo..cHi]] where

              ((_,cLo),(rHi,cHi)) = bounds f

-- Run test once; Return bool indicating success or failure. oneTest :: Int -> Int -> Double -> Rand StdGen Bool oneTest rows cols threshold =

   leaky <$> percolate <$> randomField rows cols threshold

-- Run test multple times; Return the number of tests that pass multiTest :: Int -> Int -> Int -> Double -> Rand StdGen Double multiTest repeats rows cols threshold = do

   x <- replicateM repeats $ oneTest rows cols threshold
   let leakyCount = length $ filter (==True) x
   return $ fromIntegral leakyCount / fromIntegral repeats

showField :: Field -> IO () showField a = mapM_ print [ [ a!(r,c) | c <- [cLo..cHi]] | r <- [rLo..rHi]]

             where ((rLo,cLo),(rHi,cHi)) = bounds a

main :: IO () main = do

 g <- getStdGen
 let (startField, g2) = runRand (randomField 15 15 0.6) g
 putStrLn "Unpercolated field with 0.6 threshold."
 putStrLn ""
 showField startField

 putStrLn ""
 putStrLn "Same field after percolation."
 putStrLn ""
 showField $ percolate startField

 putStrLn ""
 putStrLn "Results of running percolation test 10000 times with thresholds ranging from 0.0 to 1.0 ."
 let d = 10
 let ns = [0..10]
 let tests = sequence [multiTest 10000 15 15 v 
                          | n <- ns,
                            let v = fromIntegral n / fromIntegral d ]
 let results = zip ns (evalRand tests g2)
 mapM_ print [format ("p=" % int % "/" % int % " -> " % fixed 4) n d r | (n,r) <- results]

</lang>

Output:

Unpercolated field with 0.6 threshold.

"#### #  # #  ##"
"  # #  #       "
"#  ## # #    # "
"#   #    #  ## "
"### # #       #"
" ## #    ##### "
" ## # #     ## "
"#  # #    #  # "
"#  #  ###      "
"## ## ## ## #  "
"   ## #      ##"
"###  ## #  ####"
" ######  ##    "
"# #  #  # # # #"
" #   #  ##     "

Same field after percolation.

"####.#..#.#..##"
"  # #..#......."
"#  ##.#.#....#."
"#   #....#..##."
"### #.#.......#"
" ## #....#####."
" ## #.#.....##."
"#  # #....#..#."
"#  #  ###......"
"## ## ##.##.#.."
"   ## #......##"
"###  ##.#..####"
" ######..##    "
"# #  #..# # # #"
" #   #..##     "

Results of running percolation test 10000 times with thresholds ranging from 0.0 to 1.0 .
"p=0/10 -> 0.0000"
"p=1/10 -> 0.0000"
"p=2/10 -> 0.0000"
"p=3/10 -> 0.0000"
"p=4/10 -> 0.0030"
"p=5/10 -> 0.0908"
"p=6/10 -> 0.5648"
"p=7/10 -> 0.9557"
"p=8/10 -> 0.9996"
"p=9/10 -> 1.0000"
"p=10/10 -> 1.0000"

J

One approach:

<lang J>groups=:[: +/\ 2 </\ 0 , * ooze=: [ >. [ +&* [ * [: ; groups@[ <@(* * 2 < >./)/. + percolate=: ooze/\.@|.^:2^:_@(* (1 + # {. 1:))

trial=: percolate@([ >: ]?@$0:) simulate=: %@[ * [: +/ (2 e. {:)@trial&15 15"0@#</lang>

Example Statistics: <lang J> ,.' P THRU';(, 100&simulate)"0 (i.%<:)11 ┌────────┐ │ P THRU│ ├────────┤ │ 0 0│ │0.1 0│ │0.2 0│ │0.3 0│ │0.4 0.01│ │0.5 0.09│ │0.6 0.61│ │0.7 0.97│ │0.8 1│ │0.9 1│ │ 1 1│ └────────┘</lang>

Worked sample:

<lang J> 1j1 #"1 ' .#'{~ percolate 0.6>:?15 15$0

1. # # # # # # #
2. # # # # # # # # # # #
3. # # # # # # #

   #   # #   # # #     # # # 

1. . # # # # # #
2. # # # # # # # # #
3. # # # # # # # # # # # #

 # # # # #     #   #   # # # 

. # #

 .   .           # #   #   # 

. . . . # # # # # # # # # . . . . # # # # # # # . . . # . # # # . . . . . . . # # .

 .   . . .   . . . .   # #   </lang>

An explanation with examples would be somewhat longer than the implementation.

Alternative implementation (with an incompatible internal API):

<lang J> any =: +./ all =: *./

quickCheck =: [: all [: (any"1) 2 *./\ ] NB. a complete path requires connections between all row pairs

percolate =: 15 15&$: : (dyad define) NB. returns 0 iff blocked Use: (N, M) percolate P

NB. make a binary grid
GRID =: y (> ?@($&0)) x  

NB. compute the return value
if. -. quickCheck GRID do. 0 return. end.
STARTING_SITES =. 0 ,. ({. GRID) # i. {: x NB. indexes of 1 in head row of GRID
any STARTING_SITES check GRID

)

NB. use local copy of GRID. Too slow. check =: dyad define"1 2 NB. return 1 iff through path found use: START check GRID

GRID =. y
LOCATION =. x
if. 0 (= #) LOCATION do. 0 return. end. NB. no starting point?  0
if. LOCATION any@:((>: , 0 > [) $) GRID do. 0 return. end. NB. off grid?  0
INDEX =. <LOCATION
if. 1 ~: INDEX { GRID do. 0 return. end. NB. fail.  either already looked here or non-path
if. (>: {. LOCATION) = (# GRID) do. 1 return. end. NB. Success!  (display GRID here)
G =: GRID =. INDEX (>:@:{)`[`]}GRID
any GRID check~ LOCATION +"1 (, -)0 1,:1 0

)

NB. use global GRID. check =: dyad define"1 2 NB. return 1 iff through path found use: START check GRID

LOCATION =. x
if. 0 (= #) LOCATION do. 0 return. end. NB. no starting point?  0
if. LOCATION any@:((>: , 0 > [) $) GRID do. 0 return. end. NB. off grid?  0
INDEX =. <LOCATION
if. 1 ~: INDEX { GRID do. 0 return. end. NB. fail.  either already looked here or non-path
if. (>: {. LOCATION) = (# GRID) do. 1 return. end. NB. Success!  (display GRID here)
GRID =: INDEX (>:@:{)`[`]}GRID
any GRID check~ LOCATION +"1 (, -)0 1,:1 0

)

simulate =: 100&$: : ([ %~ [: +/ [: percolate"0 #) NB. return fraction of connected cases. Use: T simulate P </lang>

   ,. '   P  THRU' ; (, 100x&simulate)"0 (i. % <:)11x
+-----------+
|   P  THRU |
+-----------+
|   0      0|
|1r10      0|
| 1r5      0|
|3r10      0|
| 2r5  1r100|
| 1r2   1r20|
| 3r5  31r50|
|7r10 97r100|
| 4r5      1|
|9r10      1|
|   1      1|
+-----------+
   


   NB. example

   simulate 0.6
0.51

   GRID  NB. the final grid of the 100 simulated cases.
2 2 2 2 0 2 2 2 2 0 2 2 0 0 2
2 0 0 2 0 0 2 0 2 0 2 0 0 1 0
2 0 1 0 2 2 0 2 2 0 2 2 0 1 0
2 2 0 0 0 2 2 0 2 0 2 0 0 0 1
0 2 2 2 2 2 2 0 2 0 2 0 0 1 1
0 2 0 2 0 0 0 0 0 0 0 1 1 0 1
0 0 0 0 1 0 0 1 1 1 0 1 1 0 1
1 1 1 1 1 1 1 0 0 0 1 1 0 1 1
0 0 1 1 1 0 1 1 0 0 1 1 1 0 1
0 1 0 1 1 1 1 1 0 0 1 1 1 1 1
1 1 1 1 0 1 1 0 1 1 0 0 1 1 1
0 1 1 1 0 1 1 0 0 0 1 1 1 1 1
0 0 0 0 1 0 1 1 1 1 1 0 0 1 0
1 1 1 1 0 1 1 1 1 1 0 1 0 0 0
1 0 1 1 1 1 1 0 0 1 1 1 1 1 1

   
   (0 ,. 0 6 10 14) check GRID  NB. show possible starting points all fail
0 0 0 0
   
   1j1#"1 GRID { '#',~u: 32 16bb7 NB. sample paths with unicode pepper.
# # # #   # # # #   # #     #             
#     #     #   #   #     ·              
#   ·   # #   # #   # #   ·             
# #       # #   #   #       ·            
  # # # # # #   #   #     · ·           
  #   #               · ·   ·          
        ·     · · ·   · ·   ·      
· · · · · · ·       · ·   · ·  
    · · ·   · ·     · · ·   ·    
  ·   · · · · ·     · · · · ·  
· · · ·   · ·   · ·     · · ·  
  · · ·   · ·       · · · · ·   
        ·   · · · · ·     ·        
· · · ·   · · · · ·   ·         
·   · · · · ·     · · · · · · 

Python

<lang python>from random import random import string from pprint import pprint as pp

M, N, t = 15, 15, 100

cell2char = ' #' + string.ascii_letters NOT_VISITED = 1 # filled cell not walked

class PercolatedException(Exception): pass

def newgrid(p):

   return [[int(random() < p) for m in range(M)] for n in range(N)] # cell

def pgrid(cell, percolated=None):

   for n in range(N):
       print( '%i)  ' % (n % 10) 
              + ' '.join(cell2char[cell[n][m]] for m in range(M)))
   if percolated: 
       where = percolated.args[0][0]
       print('!)  ' + '  ' * where + cell2char[cell[n][where]])
   

def check_from_top(cell):

   n, walk_index = 0, 1
   try:
       for m in range(M):
           if cell[n][m] == NOT_VISITED:
               walk_index += 1
               walk_maze(m, n, cell, walk_index)
   except PercolatedException as ex:
       return ex
   return None
       

def walk_maze(m, n, cell, indx):

   # fill cell 
   cell[n][m] = indx
   # down
   if n < N - 1 and cell[n+1][m] == NOT_VISITED:
       walk_maze(m, n+1, cell, indx)
   # THE bottom
   elif n == N - 1:
       raise PercolatedException((m, indx))
   # left
   if m and cell[n][m - 1] == NOT_VISITED:
       walk_maze(m-1, n, cell, indx)
   # right
   if m < M - 1 and cell[n][m + 1] == NOT_VISITED:
       walk_maze(m+1, n, cell, indx)
   # up
   if n and cell[n-1][m] == NOT_VISITED:
       walk_maze(m, n-1, cell, indx)

if __name__ == '__main__':

   sample_printed = False
   pcount = {}
   for p10 in range(11):
       p = p10 / 10.0
       pcount[p] = 0
       for tries in range(t):
           cell = newgrid(p)
           percolated = check_from_top(cell)
           if percolated:
               pcount[p] += 1
               if not sample_printed:
                   print('\nSample percolating %i x %i, p = %5.2f grid\n' % (M, N, p))
                   pgrid(cell, percolated)
                   sample_printed = True
   print('\n p: Fraction of %i tries that percolate through\n' % t )
   
   pp({p:c/float(t) for p, c in pcount.items()})</lang>

Output:

The Ascii art grid of cells has blanks for cells that were not filled. Filled cells start off as the '#', hash character and are changed to a succession of printable characters by successive tries to navigate from the top, (top - left actually), filled cell to the bottom.

The '!)' row shows where the percolation finished and you can follow the letter backwards from that row, (letter 'c' in this case), to get the route. The program stops after finding its first route through.

Sample percolating 15 x 15, p =  0.40 grid

0)    a a a       b   c #        
1)    a a   #         c c   #   #
2)        # #   # #     c # # #  
3)  #   #       # # #   c        
4)    #     #         c c c c c c
5)  # # # # # #         c   c   c
6)        # # #         c   c   c
7)  #   #     # #     #   #   # c
8)  #   # #     #   #       c c c
9)    #       #         #   c    
0)  #       #   # # # #   c c # #
1)      #     #   #     # c      
2)  #     # # # # #   c c c   c  
3)  #   # # #         c   c c c  
4)      #           # c         #
!)                    c

 p: Fraction of 100 tries that percolate through

{0.0: 0.0,
 0.1: 0.0,
 0.2: 0.0,
 0.3: 0.0,
 0.4: 0.01,
 0.5: 0.11,
 0.6: 0.59,
 0.7: 0.94,
 0.8: 1.0,
 0.9: 1.0,
 1.0: 1.0}

Note the abrupt change in percolation at around p = 0.6. These abrupt changes are expected.

Racket

<lang racket>#lang racket (require racket/require (only-in racket/fixnum for*/fxvector)) (require (filtered-in (lambda (name) (regexp-replace #rx"unsafe-" name ""))

                     racket/unsafe/ops))

(define cell-empty 0) (define cell-filled 1) (define cell-wall 2) (define cell-visited 3) (define cell-exit 4)

(define ((percol->generator p)) (if (< (random) p) cell-filled cell-empty))

(define t (make-parameter 1000))

(define ((make-percol-grid M N) p)

 (define p->10 (percol->generator p))
 (define M+1 (fx+ 1 M))
 (define M+2 (fx+ 2 M))
 (for*/fxvector
  #:length (fx* N M+2)
  ((n (in-range N)) (m (in-range M+2)))
  (cond
    [(fx= 0 m) cell-wall]
    [(fx= m M+1) cell-wall]
    [else (p->10)])))

(define (cell->str c) (substring " #|+*" c (fx+ 1 c)))

(define ((draw-percol-grid M N) g)

 (define M+2 (fx+ M 2))
 (for ((row N))
   (for ((col (in-range M+2)))
     (define idx (fx+ (fx* M+2 row) col))
     (printf "~a" (cell->str (fxvector-ref g idx))))
   (newline)))

(define ((percolate-percol-grid?! M N) g)

 (define M+2 (fx+ M 2))
 (define N-1 (fx- N 1))
 (define max-idx (fx* N M+2))
 (define (inner-percolate g idx)
   (define row (fxquotient idx M+2))    
   (cond
     ((fx< idx 0) #f)
     ((fx>= idx max-idx) #f)
     ((fx= N-1 row) (fxvector-set! g idx cell-exit) #t)
     ((fx= cell-filled (fxvector-ref g idx))
      (fxvector-set! g idx cell-visited)
      (or 
       ; gravity first (thanks Mr Newton)
       (inner-percolate g (fx+ idx M+2))
       ; stick-to-the-left
       (inner-percolate g (fx- idx 1))
       (inner-percolate g (fx+ idx 1))
       ; go uphill only if we have to!
       (inner-percolate g (fx- idx M+2))))
     (else #f)))
 (for/first ((m (in-range 1 M+2)) #:when (inner-percolate g m)) g))

(define make-15x15-grid (make-percol-grid 15 15)) (define draw-15x15-grid (draw-percol-grid 15 15)) (define perc-15x15-grid?! (percolate-percol-grid?! 15 15))

(define (display-sample-percolation p)

 (printf "Percolation sample: p=~a~%" p)
 (for*/first
     ((i (in-naturals))
      (g (in-value (make-15x15-grid 0.6)))
      #:when (perc-15x15-grid?! g))
   (draw-15x15-grid g))
 (newline))

(display-sample-percolation 0.4)

(for ((p (sequence-map (curry * 1/10) (in-range 0 (add1 10)))))

 (define n-percolated-grids
   (for/sum
    ((i (in-range (t))) #:when (perc-15x15-grid?! (make-15x15-grid p))) 1))
 (define proportion-percolated (/ n-percolated-grids (t)))
 (printf "p=~a\t->\t~a~%" p (real->decimal-string proportion-percolated 4)))</lang>

Output:

Percolation sample: p=0.4
|+++ ++++  + +++|
| +++ ++ #     +|
|   +  ++   ##++|
| ##    +  ###+ |
| ###### + #+++#|
|  ##### +  +  #|
|## # # +++++## |
|### # ++ +++#  |
|##  ## +++++#  |
|# ###   ++ +  #|
| ## ## +++   ##|
|##  ##  +++ # #|
|###   #   +### |
|####  ####+  # |
|# ## #    *#  #|

p=0	->	0.0000
p=1/10	->	0.0000
p=1/5	->	0.0000
p=3/10	->	0.0000
p=2/5	->	0.0030
p=1/2	->	0.1110
p=3/5	->	0.5830
p=7/10	->	0.9530
p=4/5	->	1.0000
p=9/10	->	1.0000
p=1	->	1.0000

Tcl

Works with: Tcl version 8.6

<lang tcl>package require Tcl 8.6

oo::class create SitePercolation {

   variable cells w h
   constructor {width height probability} {

set w $width set h $height for {set cells {}} {[llength $cells] < $h} {lappend cells $row} { for {set row {}} {[llength $row] < $w} {lappend row $cell} { set cell [expr {rand() < $probability}] } }

   }
   method print {out} {

array set map {0 "#" 1 " " -1 .} puts "+[string repeat . $w]+" foreach row $cells { set s "|" foreach cell $row { append s $map($cell) } puts [append s "|"] } set outline [lrepeat $w "-"] foreach index $out { lset outline $index "." } puts "+[join $outline {}]+"

   }
   method percolate {} {

for {set work {}; set i 0} {$i < $w} {incr i} { if {[lindex $cells 0 $i]} {lappend work 0 $i} } try { my Fill $work return {} } trap PERCOLATED x { return [list $x] }

   }
   method Fill {queue} {

while {[llength $queue]} { set queue [lassign $queue y x] if {$y >= $h} {throw PERCOLATED $x} if {$y < 0 || $x < 0 || $x >= $w} continue if {[lindex $cells $y $x]<1} continue lset cells $y $x -1 lappend queue [expr {$y+1}] $x [expr {$y-1}] $x lappend queue $y [expr {$x-1}] $y [expr {$x+1}] }

   }

}

1. Demonstrate one run

puts "Sample percolation, 15x15 p=0.6" SitePercolation create bp 15 15 0.6 bp print [bp percolate] bp destroy puts ""

1. Collect statistics

apply {{} {

   puts "Percentage of tries that percolate, varying p"
   set tries 100
   for {set pint 0} {$pint <= 10} {incr pint} {

set p [expr {$pint * 0.1}] set tot 0 for {set i 0} {$i < $tries} {incr i} { set bp [SitePercolation new 15 15 $p] if {[$bp percolate] ne ""} { incr tot } $bp destroy } puts [format "p=%.2f: %2.1f%%" $p [expr {$tot*100./$tries}]]

   }

}}</lang>

Output:

Sample percolation, 15x15 p=0.6
+...............+
|.##...###.##...|
|.#.#####.####..|
|............##.|
|....###.###.#..|
|.#.##..#....#..|
|#.........#..#.|
|..#...##.##....|
|#.#.#....##...#|
|###.....#.#...#|
|.....##........|
|.#.#..## ......|
|  #..## # .##.#|
| # #.#  ####...|
|# #  # #  ##...|
| ###   ##  # . |
+-------------.-+

Percentage of tries that percolate, varying p
p=0.00: 0.0%
p=0.10: 0.0%
p=0.20: 0.0%
p=0.30: 0.0%
p=0.40: 0.0%
p=0.50: 6.0%
p=0.60: 54.0%
p=0.70: 98.0%
p=0.80: 100.0%
p=0.90: 100.0%
p=1.00: 100.0%

zkl

Translation of: C

<lang zkl>fcn makeGrid(m,n,p){

  grid:=Data((m+1)*(n+1));  // first row and right edges are buffers
  grid.write(" "*m); grid.write("\r");
  do(n){
     do(m){ grid.write(((0.0).random(1)<p) and "+" or "."); }  // cell is porous or not
     grid.write("\n");
  }
  grid

} fcn ff(grid,x,m){ // walk across row looking for a porous cell

  if(grid[x]!=43) return(0); // '+' == 43 ASCII == porous
  grid[x]="#";
  return(x+m>=grid.len() or 

ff(grid,x+m,m) or ff(grid,x+1,m) or ff(grid,x-1,m) or ff(grid,x-m,m)); } fcn percolate(grid,m){

  x:=m+1; i:=0; while(i<m and not ff(grid,x,m)){ x+=1; i+=1; }
  return(i<m);  // percolated through the grid?

}

grid:=makeGrid(15,15,0.60); println("Did liquid percolate: ",percolate(grid,15)); println("15x15 grid:\n",grid.text);

println("Running 10,000 tests for each case:"); foreach p in ([0.0 .. 1.0, 0.1]){

  cnt:=0.0; do(10000){ cnt+=percolate(makeGrid(15,15,p),15); }
  "p=%.1f:  %.4f".fmt(p, cnt/10000).println();

}</lang>

Output:

Did liquid percolate: True
15x15 grid:
.###.##.#++..++
......+###..+.+
+...+...##..+++
++..+.+.#+.+.++
..+++###..+..++
.+.##..++.+..++
.+#.+..++++++..
+####+..+....++
.#.#..+..++.+.+
#.#++++.+++.+++
+#++..+.+.+.+++
#######..++++++
#.##.#+++...+..
+.#.#+++.++.+++
+.+#+.++..+..++

Running 10,000 tests for each case:
p=0.0:  0.0000
p=0.1:  0.0000
p=0.2:  0.0000
p=0.3:  0.0000
p=0.4:  0.0006
p=0.5:  0.0304
p=0.6:  0.2989
p=0.7:  0.8189
p=0.8:  0.9903
p=0.9:  1.0000
p=1.0:  1.0000