Percolation/Bond percolation

Percolation/Bond 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

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. Each $\mathrm {cell} [m,n]$ is bounded by (horizontal) walls $\mathrm {hwall} [m,n]$ and $\mathrm {hwall} [m+1,n]$; (vertical) walls $\mathrm {vwall} [m,n]$ and $\mathrm {vwall} [m,n+1]$

Assume that the probability of any wall being present is a constant $p$ where

$0.0\leq p\leq 1.0$

Except for the outer horizontal walls at $m=0$ and $m=M$ which are always present.

The task

Simulate pouring a fluid onto the top surface ($n=0$) where the fluid will enter any empty cell it is adjacent to if there is no wall between where it currently is and the cell on the other side of the (missing) wall.

The fluid does not move beyond the horizontal constraints of the grid.

The fluid may move “up” within the confines of the grid of cells. If the fluid reaches a bottom cell that has a missing bottom wall then the fluid can be said to 'drip' out the bottom at that point.

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 $M=10,N=10$ grid of cells for all cases.

Optionally depict fluid successfully percolating through a grid graphically.

Show all output on this page.

C

<lang c>#include <stdio.h>

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

// cell states

1. define FILL 1
2. define RWALL 2 // right wall
3. define BWALL 4 // bottom wall

typedef unsigned int c_t;

c_t *cells, *start, *end; int m, n;

void make_grid(double p, int x, int y) {

   int i, j, thresh = RAND_MAX * p;
   m = x, n = y;

   // Allocate two addition rows to avoid checking bounds.
   // Bottom row is also required by drippage
   start = realloc(start, m * (n + 2) * sizeof(c_t));
   cells = start + m;

   for (i = 0; i < m; i++)
       start[i] = BWALL | RWALL;

   for (i = 0, end = cells; i < y; i++) {
       for (j = x; --j; )
           *end++ = (rand() < thresh ? BWALL : 0)
               |(rand() < thresh ? RWALL : 0);
       *end++ = RWALL | (rand() < thresh ? BWALL: 0);
   }
   memset(end, 0, sizeof(c_t) * m);

}

void show_grid(void) {

   int i, j;

   for (j = 0; j < m; j++) printf("+--");
   puts("+");

   for (i = 0; i <= n; i++) {
       putchar(i == n ? ' ' : '|');
       for (j = 0; j < m; j++) {
           printf((cells[i*m + j] & FILL) ? "[]" : "  ");
           putchar((cells[i*m + j] & RWALL) ? '|' : ' ');
       }
       putchar('\n');

       if (i == n) return;

       for (j = 0; j < m; j++)
           printf((cells[i*m + j] & BWALL) ? "+--" : "+  ");
       puts("+");
   }

}

int fill(c_t *p) {

   if ((*p & FILL)) return 0;
   *p |= FILL;
   if (p >= end) return 1; // success: reached bottom row

   return  ( !(p[ 0] & BWALL) && fill(p + m) ) ||
       ( !(p[ 0] & RWALL) && fill(p + 1) ) ||
       ( !(p[-1] & RWALL) && fill(p - 1) ) ||
       ( !(p[-m] & BWALL) && fill(p - m) );

}

int percolate(void) {

   int i;
   for (i = 0; i < m && !fill(cells + i); i++);

   return i < m;

}

int main(void) {

   make_grid(.5, 10, 10);
   percolate();
   show_grid();

   int cnt, i, p;

   puts("\nrunning 10x10 grids 10000 times for each p:");
   for (p = 1; p < 10; p++) {
       for (cnt = i = 0; i < 10000; i++) {
           make_grid(p / 10., 10, 10);
           cnt += percolate();
           //show_grid(); // don't
       }
       printf("p = %3g: %.4f\n", p / 10., (double)cnt / i);
   }

   free(start);
   return 0;

}</lang>

Output:

+--+--+--+--+--+--+--+--+--+--+
|[]|[] []|[] [] [] [] []      |
+  +  +  +--+--+--+--+  +  +--+
|[]|[]|[]|  |         [] []|  |
+  +--+  +--+--+--+  +--+  +  +
|[] [] [] []|        |   []|  |
+--+  +--+--+  +--+  +--+  +--+
|  |[]|     |  |        |[]   |
+--+--+  +  +  +  +  +--+  +  +
|     |  |     |         []   |
+--+  +  +--+  +--+--+  +  +--+
|  |     |     |[] [] [] []|  |
+  +  +  +--+  +  +--+--+--+--+
|  |  |     |   []   |  |  |  |
+--+  +--+--+--+  +  +  +--+  +
|  |  |  |  |  |[]|           |
+--+  +  +  +  +  +--+  +  +  +
|  |  |  |   [] []|  |  |  |  |
+--+  +--+--+  +--+  +  +  +  +
|  |     |   []|           |  |
+--+  +--+--+  +  +--+--+  +  +
             []                

running 10x10 grids 10000 times for each p:
p = 0.1: 1.0000
p = 0.2: 1.0000
p = 0.3: 0.9958
p = 0.4: 0.9123
p = 0.5: 0.5014
p = 0.6: 0.0791
p = 0.7: 0.0037
p = 0.8: 0.0000
p = 0.9: 0.0000

C++

Translation of: D

<lang cpp>#include <cstdlib>

1. include <cstring>
2. include <iostream>
3. include <string>

using namespace std;

class Grid { public:

   Grid(const double p, const int x, const int y) : m(x), n(y) {
       const int thresh = static_cast<int>(RAND_MAX * p);

       // Allocate two addition rows to avoid checking bounds.
       // Bottom row is also required by drippage
       start = new cell[m * (n + 2)];
       cells = start + m;
       for (auto i = 0; i < m; i++) start[i] = RBWALL;
       end = cells;
       for (auto i = 0; i < y; i++) {
           for (auto j = x; --j;)
               *end++ = (rand() < thresh ? BWALL : 0) | (rand() < thresh ? RWALL : 0);
           *end++ = RWALL | (rand() < thresh ? BWALL : 0);
       }
       memset(end, 0u, sizeof(cell) * m);
   }

   ~Grid() {
       delete[] start;
       cells = 0;
       start = 0;
       end = 0;
   }

   int percolate() const {
       auto i = 0;
       for (; i < m && !fill(cells + i); i++);
       return i < m;
   }

   void show() const {
       for (auto j = 0; j < m; j++)
           cout << ("+-");
       cout << '+' << endl;

       for (auto i = 0; i <= n; i++) {
           cout << (i == n ? ' ' : '|');
           for (auto j = 0; j < m; j++) {
               cout << ((cells[i * m + j] & FILL) ? "#" : " ");
               cout << ((cells[i * m + j] & RWALL) ? '|' : ' ');
           }
           cout << endl;

           if (i == n) return;

           for (auto j = 0; j < m; j++)
               cout << ((cells[i * m + j] & BWALL) ? "+-" : "+ ");
           cout << '+' << endl;
       }
   }

private:

   enum cell_state {
       FILL   = 1 << 0,
       RWALL  = 1 << 1,       // right wall
       BWALL  = 1 << 2,       // bottom wall
       RBWALL = RWALL | BWALL // right/bottom wall
   };

   typedef unsigned int cell;

   bool fill(cell* p) const {
       if ((*p & FILL)) return false;
       *p |= FILL;
       if (p >= end) return true; // success: reached bottom row

       return (!(p[0] & BWALL) && fill(p + m)) || (!(p[0] & RWALL) && fill(p + 1))
               ||(!(p[-1] & RWALL) && fill(p - 1)) || (!(p[-m] & BWALL) && fill(p - m));
   }

   cell* cells;
   cell* start;
   cell* end;
   const int m;
   const int n;

};

int main() {

   const auto M = 10, N = 10;
   const Grid grid(.5, M, N);
   grid.percolate();
   grid.show();

   const auto C = 10000;
   cout << endl << "running " << M << "x" << N << " grids " << C << " times for each p:" << endl;
   for (auto p = 1; p < M; p++) {
       auto cnt = 0, i = 0;
       for (; i < C; i++)
           cnt += Grid(p / static_cast<double>(M), M, N).percolate();
       cout << "p = " << p / static_cast<double>(M) << ": " << static_cast<double>(cnt) / i << endl;
   }

   return EXIT_SUCCESS;

}</lang>

D

Translation of: C

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

struct Grid {

   // Not enforced by runtime and type system:
   // a Cell must contain only the flags bits.
   alias Cell = uint;

   enum : Cell { // Cell states (bit flags).
       empty      = 0,
       filled     = 1,
       rightWall  = 2,
       bottomWall = 4
   }

   const size_t nc, nr;
   Cell[] cells;

   this(in size_t nRows, in size_t nCols) pure nothrow {
       nr = nRows;
       nc = nCols;

       // Allocate two addition rows to avoid checking bounds.
       // Bottom row is also required by drippage.
       cells = new Cell[nc * (nr + 2)];
   }

   void initialize(in double prob, ref Xorshift rng) {
       cells[0 .. nc] = bottomWall | rightWall; // First row.

       uint pos = nc;
       foreach (immutable r; 1 .. nr + 1) {
           foreach (immutable c; 1 .. nc)
               cells[pos++] = (uniform01 < prob ?bottomWall : empty) |
                              (uniform01 < prob ? rightWall : empty);
           cells[pos++] = rightWall |
                          (uniform01 < prob ? bottomWall : empty);
       }

       cells[$ - nc .. $] = empty; // Last row.
   }

   bool percolate() pure nothrow @nogc {
       bool fill(in size_t i) pure nothrow @nogc {
           if (cells[i] & filled)
               return false;

           cells[i] |= filled;

           if (i >= cells.length - nc)
               return true; // Success: reached bottom row.

           return (!(cells[i]      & bottomWall) && fill(i + nc)) ||
                  (!(cells[i]      & rightWall)  && fill(i + 1)) ||
                  (!(cells[i - 1]  & rightWall)  && fill(i - 1)) ||
                  (!(cells[i - nc] & bottomWall) && fill(i - nc));
       }

       return iota(nc, nc + nc).any!fill;
   }

   void show() const {
       writeln("+-".replicate(nc), '+');

       foreach (immutable r; 1 .. nr + 2) {
           write(r == nr + 1 ? ' ' : '|');
           foreach (immutable c; 0 .. nc) {
               immutable cell = cells[r * nc + c];
               write((cell & filled) ? (r <= nr ? '#' : 'X') : ' ');
               write((cell & rightWall) ? '|' : ' ');
           }
           writeln;

           if (r == nr + 1)
               return;

           foreach (immutable c; 0 .. nc)
               write((cells[r * nc + c] & bottomWall) ? "+-" : "+ ");
           '+'.writeln;
       }
   }

}

void main() {

   enum uint nr = 10, nc = 10; // N. rows and columns of the grid.
   enum uint nTries = 10_000;  // N. simulations for each probability.
   enum uint nStepsProb = 10;  // N. steps of probability.

   auto rng = Xorshift(2);
   auto g = Grid(nr, nc);
   g.initialize(0.5, rng);
   g.percolate;
   g.show;

   writefln("\nRunning %dx%d grids %d times for each p:",
            nr, nc, nTries);
   foreach (immutable p; 0 .. nStepsProb) {
       immutable probability = p / double(nStepsProb);
       uint nPercolated = 0;
       foreach (immutable i; 0 .. nTries) {
           g.initialize(probability, rng);
           nPercolated += g.percolate;
       }
       writefln("p = %0.2f: %.4f",
                probability, nPercolated / double(nTries));
   }

}</lang>

Output:

+-+-+-+-+-+-+-+-+-+-+
|#|#|#|#|     | |   |
+ +-+-+ +-+-+-+ +-+-+
|#| |  #  | | |   | |
+ +-+-+ + +-+-+ + +-+
|#|# #|#|   | |     |
+ +-+ + +-+ + + +-+ +
|#|# #|#|   | |   | |
+-+ + + + +-+-+-+-+-+
|# # # # #  | |   | |
+ + + + + + + +-+ +-+
|#|# # #|# # #  |   |
+-+ + + +-+-+ + + + +
| |#|# #| | |#      |
+-+-+-+-+ +-+ +-+-+-+
| |   |    # #|     |
+-+-+-+ +-+ +-+-+-+ +
| | |      # # #    |
+ + +-+ +-+-+-+ +-+ +
|     |   |   |#    |
+ +-+ + + + +-+ + + +
               X     

Running 10x10 grids 10000 times for each p:
p = 0.00: 1.0000
p = 0.10: 1.0000
p = 0.20: 1.0000
p = 0.30: 0.9973
p = 0.40: 0.9177
p = 0.50: 0.5050
p = 0.60: 0.0880
p = 0.70: 0.0035
p = 0.80: 0.0001
p = 0.90: 0.0000

With LDC2 compiler this code runs in 0.26 seconds (almost two times faster than the C entry).

Go

Translation of: C

<lang go>package main

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

func main() { const ( m, n = 10, 10 t = 1000 minp, maxp, Δp = 0.1, 0.99, 0.1 )

// Purposely don't seed for a 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, %.3f\n", p, float64(count)/t) } }

type cell struct { full bool right, down bool // true if open to the right (x+1) or down (y+1) }

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

func NewGrid(p float64, xsize, ysize int) *grid { g := &grid{cell: make([][]cell, ysize)} for y := range g.cell { g.cell[y] = make([]cell, xsize) for x := 0; x < xsize-1; x++ { if rand.Float64() > p { g.cell[y][x].right = true } if rand.Float64() > p { g.cell[y][x].down = true } } if rand.Float64() > p { g.cell[y][xsize-1].down = true } } return g }

var ( full = map[bool]string{false: " ", true: "**"} hopen = map[bool]string{false: "--", true: " "} vopen = map[bool]string{false: "|", true: " "} )

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) + 1) * len(g.cell[0]) * 7)

for _ = range g.cell[0] { buf.WriteString("+") buf.WriteString(hopen[false]) } buf.WriteString("+\n") for y := range g.cell { buf.WriteString(vopen[false]) for x := range g.cell[y] { buf.WriteString(full[g.cell[y][x].full]) buf.WriteString(vopen[g.cell[y][x].right]) } buf.WriteByte('\n') for x := range g.cell[y] { buf.WriteString("+") buf.WriteString(hopen[g.cell[y][x].down]) } buf.WriteString("+\n") } ly := len(g.cell) - 1 for x := range g.cell[ly] { buf.WriteByte(' ') buf.WriteString(full[g.cell[ly][x].down && g.cell[ly][x].full]) } return buf.String() }

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

func (g *grid) fill(x, y int) bool { if y >= len(g.cell) { return true // Out the bottom } if g.cell[y][x].full { return false // Allready filled } g.cell[y][x].full = true

if g.cell[y][x].down && g.fill(x, y+1) { return true } if g.cell[y][x].right && g.fill(x+1, y) { return true } if x > 0 && g.cell[y][x-1].right && g.fill(x-1, y) { return true } if y > 0 && g.cell[y-1][x].down && g.fill(x, y-1) { return true } return false }</lang>

Output:

+--+--+--+--+--+--+--+--+--+--+
|** ** **|  |  |     |  |  |  |
+  +--+  +--+--+  +--+--+--+  +
|**|  |** **|  |     |     |  |
+--+  +--+  +  +  +--+  +  +--+
|     |   **|  |              |
+--+  +--+  +--+--+--+--+--+--+
|     |   ** **|        |     |
+--+  +  +--+  +  +--+  +--+  +
|           |** ** **|     |  |
+  +  +--+  +--+  +  +--+  +--+
|  |  |  |   ** ** ** **|  |  |
+  +--+--+  +  +--+--+  +--+--+
|  |** ** **|**|**|  |** ** **|
+  +  +  +  +  +  +--+  +--+  +
|** **|**|** ** **|  |** ** **|
+  +  +--+--+--+--+  +  +--+  +
|**|** ** **|     |  |**|  |**|
+  +--+--+--+  +  +--+--+--+--+
|**               |  |  |  |  |
+  +  +  +  +--+--+  +--+--+  +
 **                           
p=0.10, 1.000
p=0.20, 1.000
p=0.30, 0.998
p=0.40, 0.915
p=0.50, 0.502
p=0.60, 0.081
p=0.70, 0.002
p=0.80, 0.000
p=0.90, 0.000

Haskell

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

data Field = Field { f :: UArray (Int, Int) Char

                  , hWall :: UArray (Int, Int) Bool  
                  , vWall :: UArray (Int, Int) Bool  
                  }

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

   let ((xLo,yLo),(xHi,yHi)) = bounds f
       validSeep = filter (\p@(x,y) ->    x >= xLo 
                                       && x <= xHi 
                                       && y >= yLo 
                                       && y <= yHi 
                                       && f!p == ' ') $ nub $ sort seep

       north (x,y) = if v ! (x  ,y  ) then [] else [(x  ,y-1)]
       south (x,y) = if v ! (x  ,y+1) then [] else [(x  ,y+1)]
       west  (x,y) = if h ! (x  ,y  ) then [] else [(x-1,y  )]
       east  (x,y) = if h ! (x+1,y  ) then [] else [(x+1,y  )]
       neighbors (x,y) = north(x,y) ++ south(x,y) ++ west(x,y) ++ east(x,y)

   in  percolateR 
           (concatMap neighbors validSeep)
           (Field (f // map (\p -> (p,'.')) validSeep) h v) 

-- Percolate a field; Return the percolated field. percolate :: Field -> Field percolate start@(Field f _ _) =

   let ((_,_),(xHi,_)) = bounds f
       (final, _) = percolateR [(x,0) | x <- [0..xHi]] start
   in final

-- Generate a random field. initField :: Int -> Int -> Double -> Rand StdGen Field initField width height threshold = do

   let f = listArray ((0,0), (width-1, height-1)) $ repeat ' '

   hrnd <- fmap (<threshold) <$> getRandoms
   let h0 = listArray ((0,0),(width, height-1)) hrnd       
       h1 = h0 // [((0,y), True) | y <- [0..height-1]]     -- close left 
       h2 = h1 // [((width,y), True) | y <- [0..height-1]] -- close right 

   vrnd <- fmap (<threshold) <$> getRandoms 
   let v0 = listArray ((0,0),(width-1, height)) vrnd   
       v1 = v0 // [((x,0), True) | x <- [0..width-1]]  -- close top

   return $ Field f h2 v1

-- Assess whether or not percolation reached bottom of field. leaks :: Field -> [Bool] leaks (Field f _ v) =

   let ((xLo,_),(xHi,yHi)) = bounds f
   in [f!(x,yHi)=='.' && not (v!(x,yHi+1)) | x <- [xLo..xHi]]

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

   or.leaks.percolate <$> initField width height threshold

-- Run test multple times; Return the number of tests that pass. multiTest :: Int -> Int -> Int -> Double -> Rand StdGen Double multiTest testCount width height threshold = do

   results <- replicateM testCount $ oneTest width height threshold
   let leakyCount = length $ filter id results
   return $ fromIntegral leakyCount / fromIntegral testCount

-- Helper function for display alternate :: [a] -> [a] -> [a] alternate [] _ = [] alternate (a:as) bs = a : alternate bs as

-- Display a field with walls and leaks. showField :: Field -> IO () showField field@(Field a h v) = do

   let ((xLo,yLo),(xHi,yHi)) = bounds a
       fLines =  [ [ a!(x,y) | x <- [xLo..xHi]] | y <- [yLo..yHi]]
       hLines =  [ [ if h!(x,y) then '|' else ' ' | x <- [xLo..xHi+1]] | y <- [yLo..yHi]]
       vLines =  [ [ if v!(x,y) then '-' else ' ' | x <- [xLo..xHi]] | y <- [yLo..yHi+1]]
       lattice =  [ [ '+' | x <- [xLo..xHi+1]] | y <- [yLo..yHi+1]]

       hDrawn = zipWith alternate hLines fLines
       vDrawn = zipWith alternate lattice vLines
   mapM_ putStrLn $ alternate vDrawn hDrawn

   let leakLine = [ if l then '.' else ' ' | l <- leaks field]
   putStrLn $ alternate (repeat ' ') leakLine

main :: IO () main = do

 g <- getStdGen
 let threshold = 0.45
     (startField, g2) = runRand (initField 10 10 threshold) g

 putStrLn ("Unpercolated field with " ++ show threshold ++ " threshold.")
 putStrLn ""
 showField startField

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

 let testCount = 10000
     densityCount = 10
 putStrLn ""
 putStrLn ("Results of running percolation test " ++ show testCount ++ " times with thresholds ranging from 0/" ++ show densityCount ++ " to " ++ show densityCount ++ "/" ++ show densityCount ++ " .")
 let densities = [0..densityCount]
 let tests = sequence [multiTest testCount 10 10 v 
                          | density <- densities,
                            let v = fromIntegral density / fromIntegral densityCount ]
 let results = zip densities (evalRand tests g2)
 mapM_ print [format ("p=" % int % "/" % int % " -> " % fixed 4) density densityCount x | (density,x) <- results]</lang>

Output:

Unpercolated field with 0.45 threshold.

+-+-+-+-+-+-+-+-+-+-+
|       | | | | | | |
+-+-+ +-+ + + + + +-+
| | |       | | | | |
+ + +-+-+ + +-+-+ + +
| |         |       |
+ +-+-+-+ +-+-+ +-+ +
| |     | |       | |
+ +-+ + + +-+-+ + +-+
| |     |       | | |
+-+-+ + + + + +-+ + +
| | |   | | | |     |
+-+ + + + + + + +-+-+
|         |   |   | |
+ + + + + +-+ +-+ + +
| | | | |     | | | |
+ + + +-+-+-+-+-+ + +
| |       |     |   |
+ +-+ +-+ +-+ + + +-+
| | | |           | |
+ + + + +-+ +-+-+-+ +
                     

Same field after percolation.

+-+-+-+-+-+-+-+-+-+-+
|. . . .|.|.|.|.|.|.|
+-+-+ +-+ + + + + +-+
| |.|. . . .|.|.|.|.|
+ + +-+-+ + +-+-+ + +
| |. . . . .|. . . .|
+ +-+-+-+ +-+-+ +-+ +
| |. . .|.|. . . .|.|
+ +-+ + + +-+-+ + +-+
| |. . .|. . . .|.|.|
+-+-+ + + + + +-+ + +
| |.|. .|.|.|.|. . .|
+-+ + + + + + + +-+-+
|. . . . .|. .|. .|.|
+ + + + + +-+ +-+ + +
|.|.|.|.|. . .| |.|.|
+ + + +-+-+-+-+-+ + +
|.|. . . .|. . .|. .|
+ +-+ +-+ +-+ + + +-+
|.| |.|. . . . . .| |
+ + + + +-+ +-+-+-+ +
 .   . .   .         

Results of running percolation test 10000 times with thresholds ranging from 0/10 to 10/10 .
"p=0/10 -> 1.0000"
"p=1/10 -> 1.0000"
"p=2/10 -> 1.0000"
"p=3/10 -> 0.9969"
"p=4/10 -> 0.9171"
"p=5/10 -> 0.5026"
"p=6/10 -> 0.0901"
"p=7/10 -> 0.0025"
"p=8/10 -> 0.0000"
"p=9/10 -> 0.0000"
"p=10/10 -> 0.0000"

Perl 6

Works with: Rakudo version 2017.02

Starts "filling" from the top left. Fluid flow favours directions in Down, Left, Right, Up order. I interpreted p to be porosity, so small p mean low permeability, large p means high permeability.

<lang perl6>my @bond; my $grid = 10; my $geom = $grid - 1; my $water = '▒';

enum Direction <DeadEnd Up Right Down Left>;

say 'Sample percolation at .6'; percolate .6; .join.say for @bond; say "\n";

my $tests = 100; say "Doing $tests trials at each porosity:"; for .1, .2 ... 1 -> $p {

   printf "p = %0.1f: %0.2f\n", $p, (sum percolate($p) xx $tests) / $tests

}

sub percolate ( $prob ) {

   generate $prob;
   my @stack;
   my $current = [1;0];
   $current.&fill;

   loop {
       if my $dir = direction( $current ) {
           @stack.push: $current;
           $current = move $dir, $current
       }
       else {
           return 0 unless @stack;
           $current = @stack.pop
       }
       return 1 if $current[1] == +@bond - 1
   }

   sub direction( [$x, $y] ) {
       ( Down  if @bond[$y + 1][$x].contains: ' ' ) ||
       ( Left  if @bond[$y][$x - 1].contains: ' ' ) ||
       ( Right if @bond[$y][$x + 1].contains: ' ' ) ||
       ( Up    if @bond[$y - 1][$x].defined && @bond[$y - 1][$x].contains: ' ' ) ||
       DeadEnd
   }

   sub move ( $dir, @cur ) {
       my ( $x, $y ) = @cur;
       given $dir {
           when Up    { [$x,--$y].&fill xx 2 }
           when Down  { [$x,++$y].&fill xx 2 }
           when Left  { [--$x,$y].&fill xx 2 }
           when Right { [++$x,$y].&fill xx 2 }
       }
       [$x, $y]
   }

   sub fill ( [$x, $y] ) { @bond[$y;$x].=subst(' ', $water, :g) }

}

sub generate ( $prob = .5 ) {

   @bond = ();
   my $sp = '   ';
   append @bond, [flat '│', ($sp, ' ') xx $geom, $sp, '│'],
                 [flat '├', (h(), '┬') xx $geom, h(), '┤'];
   append @bond, [flat '│', ($sp, v()) xx $geom, $sp, '│'],
                 [flat '├', (h(), '┼') xx $geom, h(), '┤'] for ^$geom;
   append @bond, [flat '│', ($sp, v()) xx $geom, $sp, '│'],
                 [flat '├', (h(), '┴') xx $geom, h(), '┤'],
                 [flat '│', ($sp, ' ') xx $geom, $sp, '│'];

   sub h () { rand < $prob ?? $sp !! '───' }
   sub v () { rand < $prob ?? ' ' !! '│'   }

}</lang>

Output:

Sample percolation at .6
│▒▒▒                                    │
├▒▒▒┬   ┬───┬   ┬   ┬   ┬   ┬   ┬───┬   ┤
│▒▒▒▒▒▒▒                │   │           │
├───┼▒▒▒┼   ┼   ┼   ┼   ┼   ┼───┼   ┼   ┤
│▒▒▒▒▒▒▒▒▒▒▒│   │   │   │   │   │       │
├▒▒▒┼───┼▒▒▒┼   ┼───┼   ┼───┼   ┼   ┼   ┤
│▒▒▒│▒▒▒▒▒▒▒▒▒▒▒                │   │   │
├▒▒▒┼───┼───┼▒▒▒┼   ┼   ┼───┼   ┼   ┼   ┤
│▒▒▒│        ▒▒▒│   │   │           │   │
├───┼   ┼   ┼▒▒▒┼───┼   ┼   ┼   ┼   ┼───┤
│           │▒▒▒    │                   │
├   ┼───┼   ┼▒▒▒┼───┼───┼───┼───┼   ┼───┤
│           │▒▒▒│                       │
├───┼   ┼───┼▒▒▒┼───┼───┼───┼───┼   ┼───┤
│▒▒▒▒▒▒▒▒▒▒▒▒▒▒▒        │       │       │
├▒▒▒┼▒▒▒┼───┼▒▒▒┼───┼   ┼───┼   ┼   ┼   ┤
│▒▒▒│▒▒▒▒▒▒▒│▒▒▒▒▒▒▒│                   │
├▒▒▒┼───┼───┼───┼───┼───┼   ┼   ┼   ┼   ┤
│▒▒▒▒▒▒▒    │       │   │               │
├▒▒▒┼▒▒▒┼───┼───┼   ┼───┼───┼   ┼   ┼   ┤
│▒▒▒│▒▒▒    │               │           │
├───┴▒▒▒┴   ┴   ┴   ┴───┴   ┴   ┴   ┴───┤
│    ▒▒▒                                │


Doing 100 trials at each porosity:
p = 0.1: 0.00
p = 0.2: 0.00
p = 0.3: 0.00
p = 0.4: 0.05
p = 0.5: 0.42
p = 0.6: 0.92
p = 0.7: 1.00
p = 0.8: 1.00
p = 0.9: 1.00
p = 1.0: 1.00

Python

<lang python>from collections import namedtuple from random import random from pprint import pprint as pp

Grid = namedtuple('Grid', 'cell, hwall, vwall')

M, N, t = 10, 10, 100

class PercolatedException(Exception): pass

HVF = [(' .', ' _'), (':', '|'), (' ', '#')] # Horiz, vert, fill chars

def newgrid(p):

   hwall = [[int(random() < p) for m in range(M)] 
            for n in range(N+1)]
   vwall = [[(1 if m in (0, M) else int(random() < p)) for m in range(M+1)] 
            for n in range(N)]
   cell = [[0 for m in range(M)] 
            for n in range(N)]
   return Grid(cell, hwall, vwall)

def pgrid(grid, percolated=None):

   cell, hwall, vwall = grid
   h, v, f = HVF
   for n in range(N):
       print('    ' + .join(h[hwall[n][m]] for m in range(M)))
       print('%i)  ' % (n % 10) + .join(v[vwall[n][m]] + f[cell[n][m] if m < M else 0]
                                         for m in range(M+1))[:-1])
   n = N
   print('    ' + .join(h[hwall[n][m]] for m in range(M)))
   if percolated: 
       where = percolated.args[0][0]
       print('!)  ' + '  ' * where + ' ' + f[1])
   

def pour_on_top(grid):

   cell, hwall, vwall = grid
   n = 0
   try:
       for m in range(M):
           if not hwall[n][m]:
               flood_fill(m, n, cell, hwall, vwall)
   except PercolatedException as ex:
       return ex
   return None
       

def flood_fill(m, n, cell, hwall, vwall):

   # fill cell 
   cell[n][m] = 1
   # bottom
   if n < N - 1 and not hwall[n + 1][m] and not cell[n+1][m]:
       flood_fill(m, n+1, cell, hwall, vwall)
   # THE bottom
   elif n == N - 1 and not hwall[n + 1][m]:
       raise PercolatedException((m, n+1))
   # left
   if m and not vwall[n][m] and not cell[n][m - 1]:
       flood_fill(m-1, n, cell, hwall, vwall)
   # right
   if m < M - 1 and not vwall[n][m + 1] and not cell[n][m + 1]:
       flood_fill(m+1, n, cell, hwall, vwall)
   # top
   if n and not hwall[n][m] and not cell[n-1][m]:
       flood_fill(m, n-1, cell, hwall, vwall)

if __name__ == '__main__':

   sample_printed = False
   pcount = {}
   for p10 in range(11):
       p = (10 - p10) / 10.0    # count down so sample print is interesting
       pcount[p] = 0
       for tries in range(t):
           grid = newgrid(p)
           percolated = pour_on_top(grid)
           if percolated:
               pcount[p] += 1
               if not sample_printed:
                   print('\nSample percolating %i x %i grid' % (M, N))
                   pgrid(grid, percolated)
                   sample_printed = True
   print('\n p: Fraction of %i tries that percolate through' % t )
   
   pp({p:c/float(t) for p, c in pcount.items()})</lang>

Output:

In the Ascii art, cells are either a space or a hash and are surrounded by either '_', '|' for intact walls and '.' and ':' for missing (leaky) walls.

The bottom-most line starting '!)' shows where the fluid can drip out from. (The percolation stops when one route through the bottom is found).

Sample percolating 10 x 10 grid
     _ _ . _ . _ _ . _ _
0)  | |#:#:#|#| | :#| | |
     _ _ . _ _ _ . . _ _
1)  | | |#:#| | | |#| : |
     _ _ _ . _ . . . . _
2)  | | |#:#| : | |#: | |
     _ _ _ _ . . _ . . .
3)  | : : | | | : |#: | |
     _ _ . _ . . _ . _ _
4)  | : : : | | | |#: : |
     _ _ _ . _ _ _ . . _
5)  | : | | : | | :#| | |
     _ _ . . _ _ _ . _ .
6)  | : | | : | |#:#:#| |
     _ . _ _ . _ _ _ . .
7)  | : | : | : | | |#: |
     _ _ _ . . _ _ . . _
8)  | | : | | | |#:#:#: |
     _ _ _ . . . . _ _ .
9)  | : : | : : :#: | : |
     . _ . _ . . . . _ _
!)               #

 p: Fraction of 100 tries that percolate through
{0.0: 1.0,
 0.1: 1.0,
 0.2: 1.0,
 0.3: 1.0,
 0.4: 0.9,
 0.5: 0.47,
 0.6: 0.06,
 0.7: 0.0,
 0.8: 0.0,
 0.9: 0.0,
 1.0: 0.0}

Note the abrupt cut-off in percolation at around p = 0.5 which is to be expected.

Racket

<lang racket>#lang racket

(define has-left-wall? (lambda (x) (bitwise-bit-set? x 0))) (define has-right-wall? (lambda (x) (bitwise-bit-set? x 1))) (define has-top-wall? (lambda (x) (bitwise-bit-set? x 2))) (define has-bottom-wall? (lambda (x) (bitwise-bit-set? x 3))) (define has-fluid? (lambda (x) (bitwise-bit-set? x 4)))

(define (walls->cell l? r? t? b?)

 (+ (if l? 1 0) (if r? 2 0) (if t? 4 0) (if b? 8 0)))

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

 (define rv (make-vector (* M N)))
 (for* ((idx (in-range (* M N))))
   (define left-wall?
     (or (zero? (modulo idx M))
         (has-right-wall? (vector-ref rv (sub1 idx)))))
   (define right-wall?
     (or (= (modulo idx M) (sub1 M))
         (< (random) p)))
   (define top-wall?
     (if (< idx M) (< (random) p)
         (has-bottom-wall? (vector-ref rv (- idx M)))))
   (define bottom-wall? (< (random) p))    
   (define cell-value
     (walls->cell left-wall? right-wall? top-wall? bottom-wall?))
   (vector-set! rv idx cell-value))
 rv)

(define (display-percol-grid M . vs)

 (define N (/ (vector-length (car vs)) M))
 (define-syntax-rule (tab-eol m)
   (when (= m (sub1 M)) (printf "\t")))
 (for ((n N))
   (for* ((v vs) (m M))
     (when (zero? m) (printf "+"))
     (printf 
      (match (vector-ref v (+ (* n M) m))
        ((? has-top-wall?) "-+")
        ((? has-fluid?)    "#+")
        (else ".+")))
     (tab-eol m))
   (newline)
   (for* ((v vs) (m M))
     (when (zero? m) (printf "|"))
     (printf
      (match (vector-ref v (+ (* n M) m))
        ((and (? has-fluid?) (? has-right-wall?)) "#|")
        ((? has-right-wall?) ".|")
        ((? has-fluid?) "##")
        (else "..")))
     (tab-eol m))
   (newline))
 (for* ((v vs) (m M))
   (when (zero? m) (printf "+"))
   (printf 
    (match (vector-ref v (+ (* (sub1 M) M) m))
      ((? has-bottom-wall?) "-+")
      ((? has-fluid?)    "#+")
      (else ".+")))
   (tab-eol m))
 (newline))

(define (find-bonded-grid-t/b-path M v)

 (define N (/ (vector-length v) M))
 
 (define (flood-cell idx)
   (cond
     [(= (quotient idx M) N) #t] ; wootiments!
     [(has-fluid? (vector-ref v idx)) #f] ; been here
     [else (define cell (vector-ref v idx))
           (vector-set! v idx (bitwise-ior cell 16))                     
           (or (and (not (has-bottom-wall? cell)) (flood-cell (+ idx M)))
               (and (not (has-left-wall? cell))   (flood-cell (- idx 1)))
               (and (not (has-right-wall? cell))  (flood-cell (+ idx 1)))
               (and (not (has-top-wall? cell))
                    (>= idx M) ; not top row
                    (flood-cell (- idx M))))]))
 
 (for/first ((m (in-range M))
             #:unless (has-top-wall? (vector-ref v m))
             #:when (flood-cell m)) #t))

(define t (make-parameter 1000)) (define (experiment p)

 (/ (for*/sum ((sample (in-range (t)))
               (v (in-value (bonded-percol-grid 10 10 p)))
               #:when (find-bonded-grid-t/b-path 10 v)) 1)
    (t)))

(define (main)

 (for ((tenths (in-range 0 (add1 10))))
   (define p (/ tenths 10))
   (define e (experiment p))
   (printf "proportion of grids that percolate p=~a : ~a (~a)~%"
           p e (real->decimal-string e 5))))

(module+ test

 (define (make/display/flood/display-bonded-grid M N p attempts (atmpt 1))
   (define v (bonded-percol-grid M N p))
   (define v+ (vector-copy v))
   (cond [(or (find-bonded-grid-t/b-path M v+) (= attempts 0))
          (define v* (vector-copy v+))
          (define (flood-bonded-grid)
            (when (find-bonded-grid-t/b-path M v*)
              (flood-bonded-grid)))
          (flood-bonded-grid)
          (display-percol-grid M v v+ v*)
          (printf "After ~a attempt(s)~%~%" atmpt)]
         [else
          (make/display/flood/display-bonded-grid
           M N p (sub1 attempts) (add1 atmpt))]))
 
 (make/display/flood/display-bonded-grid 10 10 0   20)
 (make/display/flood/display-bonded-grid 10 10 .25 20)
 (make/display/flood/display-bonded-grid 10 10 .50 20)
 (make/display/flood/display-bonded-grid 10 10 .75 20000))</lang>

Output:

Welcome to DrRacket, version 5.3.5 [3m].
Language: racket [custom]; memory limit: 1024 MB.
+.+.+.+.+.+.+.+.+.+.+	+#+.+.+.+.+.+.+.+.+.+	+#+#+#+#+#+#+#+#+#+#+	
|...................|	|##.................|	|###################|	
+.+.+.+.+.+.+.+.+.+.+	+#+.+.+.+.+.+.+.+.+.+	+#+#+#+#+#+#+#+#+#+#+	
|...................|	|##.................|	|###################|	
+.+.+.+.+.+.+.+.+.+.+	+#+.+.+.+.+.+.+.+.+.+	+#+#+#+#+#+#+#+#+#+#+	
|...................|	|##.................|	|###################|	
+.+.+.+.+.+.+.+.+.+.+	+#+.+.+.+.+.+.+.+.+.+	+#+#+#+#+#+#+#+#+#+#+	
|...................|	|##.................|	|###################|	
+.+.+.+.+.+.+.+.+.+.+	+#+.+.+.+.+.+.+.+.+.+	+#+#+#+#+#+#+#+#+#+#+	
|...................|	|##.................|	|###################|	
+.+.+.+.+.+.+.+.+.+.+	+#+.+.+.+.+.+.+.+.+.+	+#+#+#+#+#+#+#+#+#+#+	
|...................|	|##.................|	|###################|	
+.+.+.+.+.+.+.+.+.+.+	+#+.+.+.+.+.+.+.+.+.+	+#+#+#+#+#+#+#+#+#+#+	
|...................|	|##.................|	|###################|	
+.+.+.+.+.+.+.+.+.+.+	+#+.+.+.+.+.+.+.+.+.+	+#+#+#+#+#+#+#+#+#+#+	
|...................|	|##.................|	|###################|	
+.+.+.+.+.+.+.+.+.+.+	+#+.+.+.+.+.+.+.+.+.+	+#+#+#+#+#+#+#+#+#+#+	
|...................|	|##.................|	|###################|	
+.+.+.+.+.+.+.+.+.+.+	+#+.+.+.+.+.+.+.+.+.+	+#+#+#+#+#+#+#+#+#+#+	
|...................|	|##.................|	|###################|	
+.+.+.+.+.+.+.+.+.+.+	+#+.+.+.+.+.+.+.+.+.+	+#+#+#+#+#+#+#+#+#+#+	
After 1 attempt(s)

+.+-+-+.+.+.+-+.+.+.+	+#+-+-+.+.+.+-+.+.+.+	+#+-+-+#+#+#+-+#+#+#+	
|...................|	|##.................|	|##..###############|	
+.+-+.+-+.+.+-+-+-+.+	+#+-+.+-+.+.+-+-+-+.+	+#+-+#+-+#+#+-+-+-+#+	
|.................|.|	|##...............|.|	|##..##..####.....|#|	
+.+-+.+.+.+.+-+.+.+.+	+#+-+.+.+.+.+-+.+.+.+	+#+-+#+.+#+#+-+.+.+#+	
|.............|.....|	|##...........|.....|	|######..#####|....#|	
+.+.+.+.+.+.+.+.+.+.+	+#+.+.+.+.+.+.+.+.+.+	+#+#+#+.+#+#+#+.+.+#+	
|.....|...|.|.......|	|##...|...|.|.......|	|#####|..#|#|##....#|	
+.+.+.+.+.+.+.+-+-+.+	+#+.+.+.+.+.+.+-+-+.+	+#+#+#+#+#+#+#+-+-+#+	
|.|.............|...|	|#|.............|...|	|#|############.|..#|	
+.+-+-+.+-+.+.+.+.+.+	+#+-+-+.+-+.+.+.+.+.+	+#+-+-+#+-+#+#+.+.+#+	
|...................|	|##.................|	|##....##..####....#|	
+.+.+-+.+.+.+.+-+-+.+	+#+.+-+.+.+.+.+-+-+.+	+#+.+-+#+.+#+#+-+-+#+	
|...|...|...........|	|##.|...|...........|	|##.|###|..####..###|	
+.+.+.+-+.+.+.+.+.+.+	+#+#+.+-+.+.+.+.+.+.+	+#+#+#+-+.+#+#+.+#+#+	
|...|...|.........|.|	|###|...|.........|.|	|###|##.|..####..#|#|	
+-+.+.+-+-+.+.+.+.+-+	+-+#+.+-+-+.+.+.+.+-+	+-+#+#+-+-+#+#+.+#+-+	
|.....|.........|...|	|..##.|.........|...|	|..###|....####.|###|	
+.+.+.+.+.+.+.+.+.+.+	+.+#+.+.+.+.+.+.+.+.+	+.+#+#+.+.+#+#+#+#+#+	
|.........|.......|.|	|..##.....|.......|.|	|..####...|#######|#|	
+.+.+.+-+.+.+-+.+-+.+	+.+#+.+-+.+.+-+.+-+.+	+.+#+#+-+.+#+-+#+-+#+	
After 1 attempt(s)

+.+.+.+.+-+-+.+-+.+.+	+#+#+#+#+-+-+.+-+.+.+	+#+#+#+#+-+-+#+-+#+#+	
|.........|.|.|...|.|	|########.|.|.|...|.|	|########.|.|#|###|#|	
+.+-+-+.+-+-+-+.+.+-+	+#+-+-+#+-+-+-+.+.+-+	+#+-+-+#+-+-+-+#+#+-+	
|...|...|...|.|.|.|.|	|###|..#|...|.|.|.|.|	|###|..#|...|.|#|#|.|	
+-+-+.+.+.+.+-+.+-+.+	+-+-+.+#+#+.+-+.+-+.+	+-+-+.+#+#+.+-+#+-+.+	
|.|.|.|...|.|.|.|...|	|.|.|.|###|.|.|.|...|	|.|.|.|###|.|.|#|...|	
+.+-+.+-+.+.+.+-+.+-+	+.+-+.+-+#+.+.+-+.+-+	+.+-+.+-+#+.+.+-+.+-+	
|.|...|...|.|.....|.|	|.|...|###|.|.....|.|	|.|...|###|.|.....|.|	
+.+-+.+.+.+-+-+.+.+.+	+.+-+.+#+#+-+-+.+.+.+	+.+-+.+#+#+-+-+.+.+.+	
|.|...|.|.....|.....|	|.|...|#|####.|.....|	|.|...|#|####.|.....|	
+-+.+-+.+-+.+-+.+-+-+	+-+.+-+#+-+#+-+#+-+-+	+-+.+-+#+-+#+-+#+-+-+	
|.|.|.....|.....|...|	|.|.|#####|#####|...|	|.|.|#####|#####|...|	
+-+-+.+.+.+.+-+.+-+-+	+-+-+#+#+#+#+-+#+-+-+	+-+-+#+#+#+#+-+#+-+-+	
|...|.|.....|.......|	|...|#|#####|..##...|	|...|#|#####|..##...|	
+-+-+-+-+-+-+-+.+-+-+	+-+-+-+-+-+-+-+#+-+-+	+-+-+-+-+-+-+-+#+-+-+	
|.|...|.|.|.......|.|	|.|...|.|.|######.|.|	|.|...|.|.|######.|.|	
+.+-+-+-+.+.+-+.+.+.+	+.+-+-+-+.+#+-+#+.+.+	+.+-+-+-+.+#+-+#+.+.+	
|.|...|.......|.|.|.|	|.|...|....##.|#|.|.|	|.|...|....##.|#|.|.|	
+.+.+-+.+.+.+-+-+-+-+	+.+.+-+.+.+#+-+-+-+-+	+.+.+-+.+.+#+-+-+-+-+	
|.|.........|.....|.|	|.|........#|.....|.|	|.|........#|.....|.|	
+-+.+-+-+-+.+.+.+-+.+	+-+.+-+-+-+#+.+.+-+.+	+-+.+-+-+-+#+.+.+-+.+	
After 2 attempt(s)

+-+-+-+-+-+-+.+-+-+.+	+-+-+-+-+-+-+#+-+-+.+	+-+-+-+-+-+-+#+-+-+#+	
|.|.|...|.|.|.|.|...|	|.|.|...|.|.|#|.|...|	|.|.|...|.|.|#|.|###|	
+-+-+-+-+-+-+.+-+-+-+	+-+-+-+-+-+-+#+-+-+-+	+-+-+-+-+-+-+#+-+-+-+	
|.|.|.|...|.|...|.|.|	|.|.|.|...|.|##.|.|.|	|.|.|.|...|.|##.|.|.|	
+.+.+.+.+.+-+.+-+-+.+	+.+.+.+.+.+-+#+-+-+.+	+.+.+.+.+.+-+#+-+-+.+	
|.|.|.|.|...|.|...|.|	|.|.|.|.|...|#|...|.|	|.|.|.|.|...|#|...|.|	
+.+-+.+-+-+-+.+-+.+.+	+.+-+.+-+-+-+#+-+.+.+	+.+-+.+-+-+-+#+-+.+.+	
|...|...|.|.|...|.|.|	|...|...|.|.|###|.|.|	|...|...|.|.|###|.|.|	
+-+-+-+-+-+-+-+.+-+-+	+-+-+-+-+-+-+-+#+-+-+	+-+-+-+-+-+-+-+#+-+-+	
|.|.......|.....|.|.|	|.|.......|#####|.|.|	|.|.......|#####|.|.|	
+.+-+-+-+.+.+-+-+-+-+	+.+-+-+-+.+#+-+-+-+-+	+.+-+-+-+.+#+-+-+-+-+	
|.|.|.|.|.|.|.|.....|	|.|.|.|.|.|#|.|.....|	|.|.|.|.|.|#|.|.....|	
+-+-+-+-+-+.+.+-+.+-+	+-+-+-+-+-+#+.+-+.+-+	+-+-+-+-+-+#+.+-+.+-+	
|...|.|.|.|.|.|.|.|.|	|...|.|.|.|#|.|.|.|.|	|...|.|.|.|#|.|.|.|.|	
+.+.+.+-+-+.+.+-+.+-+	+.+.+.+-+-+#+#+-+.+-+	+.+.+.+-+-+#+#+-+.+-+	
|.|.|.|.|.|...|.|...|	|.|.|.|.|.|###|.|...|	|.|.|.|.|.|###|.|...|	
+-+-+-+-+-+-+.+.+.+-+	+-+-+-+-+-+-+#+.+.+-+	+-+-+-+-+-+-+#+.+.+-+	
|.|.|.|.|.|...|...|.|	|.|.|.|.|.|###|...|.|	|.|.|.|.|.|###|...|.|	
+-+-+.+-+-+.+-+-+-+.+	+-+-+.+-+-+#+-+-+-+.+	+-+-+.+-+-+#+-+-+-+.+	
|.|.|.|...|...|.|...|	|.|.|.|...|###|.|...|	|.|.|.|...|###|.|...|	
+-+-+.+-+.+-+.+.+.+-+	+-+-+.+-+.+-+#+.+.+-+	+-+-+.+-+.+-+#+.+.+-+	
After 4611 attempt(s)

> (main)
proportion of grids that percolate p=0 : 1 (1.00000)
proportion of grids that percolate p=1/10 : 1 (1.00000)
proportion of grids that percolate p=1/5 : 1 (1.00000)
proportion of grids that percolate p=3/10 : 199/200 (0.99500)
proportion of grids that percolate p=2/5 : 179/200 (0.89500)
proportion of grids that percolate p=1/2 : 451/1000 (0.45100)
proportion of grids that percolate p=3/5 : 29/500 (0.05800)
proportion of grids that percolate p=7/10 : 1/1000 (0.00100)
proportion of grids that percolate p=4/5 : 0 (0.00000)
proportion of grids that percolate p=9/10 : 0 (0.00000)
proportion of grids that percolate p=1 : 0 (0.00000)

Tcl

Works with: Tcl version 8.6

Translation of: Python

<lang tcl>package require Tcl 8.6

1. Structure the bond percolation system as a class

oo::class create BondPercolation {

   variable hwall vwall cells M N
   constructor {width height probability} {

set M $height set N $width for {set i 0} {$i <= $height} {incr i} { for {set j 0;set walls {}} {$j < $width} {incr j} { lappend walls [expr {rand() < $probability}] } lappend hwall $walls } for {set i 0} {$i <= $height} {incr i} { for {set j 0;set walls {}} {$j <= $width} {incr j} { lappend walls [expr {$j==0 || $j==$width || rand() < $probability}] } lappend vwall $walls } set cells [lrepeat $height [lrepeat $width 0]]

   }

   method print Template:Percolated "" {

set nw [string length $M] set grid $cells if {$percolated ne ""} { lappend grid [lrepeat $N 0] lset grid end $percolated 1 } foreach hws $hwall vws [lrange $vwall 0 end-1] r $grid { incr row puts -nonewline [string repeat " " [expr {$nw+2}]] foreach w $hws { puts -nonewline [if {$w} {subst "+-"} {subst "+ "}] } puts "+" puts -nonewline [format "%-*s" [expr {$nw+2}] [expr { $row>$M ? $percolated eq "" ? " " : ">" : "$row)" }]] foreach v $vws c $r { puts -nonewline [if {$v==1} {subst "|"} {subst " "}] puts -nonewline [if {$c==1} {subst "#"} {subst " "}] } puts "" }

   }

   method percolate {} {

try { for {set i 0} {$i < $N} {incr i} { if {![lindex $hwall 0 $i]} { my FloodFill $i 0 } } return "" } trap PERCOLATED n { return $n }

   }
   method FloodFill {x y} {

# fill cell lset cells $y $x 1 # bottom if {![lindex $hwall [expr {$y+1}] $x]} { if {$y == $N-1} { # THE bottom throw PERCOLATED $x } if {$y < $N-1 && ![lindex $cells [expr {$y+1}] $x]} { my FloodFill $x [expr {$y+1}] } } # left if {![lindex $vwall $y $x] && ![lindex $cells $y [expr {$x-1}]]} { my FloodFill [expr {$x-1}] $y } # right if {![lindex $vwall $y [expr {$x+1}]] && ![lindex $cells $y [expr {$x+1}]]} { my FloodFill [expr {$x+1}] $y } # top if {$y>0 && ![lindex $hwall $y $x] && ![lindex $cells [expr {$y-1}] $x]} { my FloodFill $x [expr {$y-1}] }

   }

}

1. Demonstrate one run

puts "Sample percolation, 10x10 p=0.5" BondPercolation create bp 10 10 0.5 bp print [bp percolate] bp destroy puts ""

1. Collect some aggregate 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 [BondPercolation new 10 10 $p] if {[$bp percolate] ne ""} { incr tot } $bp destroy } puts [format "p=%.2f: %2.1f%%" $p [expr {$tot*100./$tries}]]

   }

}}</lang>

Output:

Sample percolation, 10x10 p=0.5
    + + +-+-+-+ +-+ +-+ +
1)  |#  |   |   |   |   | 
    + +-+ + + +-+ + + +-+
2)  |#|       | |     | | 
    + + +-+ +-+ +-+ + +-+
3)  |# # #|# # #| | |   | 
    + +-+ + +-+ +-+ +-+ +
4)  |#|# # #| |#  |     | 
    +-+ + + +-+ +-+-+ +-+
5)  |# # # #| |#  |   | | 
    +-+-+-+-+ + + + +-+-+
6)  | |     | |#|   |   | 
    +-+-+-+-+-+ + +-+-+ +
7)  | | | |   |#      | | 
    + +-+ +-+-+ +-+ +-+ +
8)  |       |  #    |   | 
    + +-+-+ +-+ + + + + +
9)  |          #        | 
    + + +-+-+ + +-+-+ + +
10) |   | |    #  | |   | 
    + + + + + + +-+ +-+ +
>              #        

Percentage of tries that percolate, varying p
p=0.00: 100.0%
p=0.10: 100.0%
p=0.20: 100.0%
p=0.30: 100.0%
p=0.40: 86.0%
p=0.50: 50.0%
p=0.60: 6.0%
p=0.70: 0.0%
p=0.80: 0.0%
p=0.90: 0.0%
p=1.00: 0.0%

zkl

Translation of: C

<lang zkl>// cell states const FILLED=1; // and odd const RWALL =2; // right wall const BWALL =4; // bottom wall

fcn P(p,wall){ (0.0).random(1)

=grid.len()) return(True); // success: reached bottom row return(( not grid[x] .bitAnd(BWALL) and fill(grid,x + m,m) ) or // down ( not grid[x] .bitAnd(RWALL) and fill(grid,x + 1,m) ) or // right ( not grid[x - 1].bitAnd(RWALL) and fill(grid,x - 1,m) ) or // left ( not grid[x - m].bitAnd(BWALL) and fill(grid,x - m,m) )); // up } fcn percolate(grid,m){ i:=0; while(i<m and not fill(grid,i+m,m)){ i+=1; } // pour juice on top row return(i<m); // percolated through the grid? }</lang> <lang zkl>grid:=makeGrid(10,10,0.40); println("Did liquid percolate: ",percolate(grid,10)); show(grid,10,10); 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(10,10,p),10); } "p=%.1f:  %.4f".fmt(p, cnt/10000).println(); }</lang>

Output:

Did liquid percolate: True
+--+--+--+--+--+--+--+--+--+--+
|** **      |              |  |
+--+  +--+--+  +  +  +  +  +  +
|   **|  |        |        |  |
+  +  +  +--+  +--+--+  +--+--+
|   ** **      |              |
+--+--+  +  +  +  +--+  +  +--+
|     |**|  |        |        |
+  +  +  +  +--+  +  +--+  +  +
|      ** **|  |** **|     |  |
+  +--+--+  +--+  +  +--+  +  +
|     |  |**|  |**|** **      |
+  +  +  +  +--+  +--+  +  +  +
|     |  |** ** ** **|**      |
+--+--+--+--+  +--+--+  +--+--+
|  |     |** **|      **   |  |
+  +  +--+  +  +  +  +  +--+--+
|        |** **|     |**|  |  |
+  +--+  +--+--+--+--+  +  +  +
|              |  |   **|     |
+  +  +  +  +  +  +--+  +  +  +
                      **       
Running 10,000 tests for each case:
p=0.0:  1.0000
p=0.1:  1.0000
p=0.2:  1.0000
p=0.3:  0.9978
p=0.4:  0.9163
p=0.5:  0.5017
p=0.6:  0.0890
p=0.7:  0.0033
p=0.8:  0.0000
p=0.9:  0.0000
p=1.0:  0.0000