# Percolation/Mean cluster density

Percolation/Mean cluster density
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

Let ${\displaystyle c}$ be a 2D boolean square matrix of ${\displaystyle n\times n}$ values of either 1 or 0 where the probability of any value being 1 is ${\displaystyle p}$, (and of 0 is therefore ${\displaystyle 1-p}$). We define a cluster of 1's as being a group of 1's connected vertically or horizontally (i.e., using the Von Neumann neighborhood rule) and bounded by either ${\displaystyle 0}$ or by the limits of the matrix. Let the number of such clusters in such a randomly constructed matrix be ${\displaystyle C_{n}}$.

Percolation theory states that ${\displaystyle K(p)}$ (the mean cluster density) will satisfy ${\displaystyle K(p)=C_{n}/n^{2}}$ as ${\displaystyle n}$ tends to infinity. For ${\displaystyle p=0.5}$, ${\displaystyle K(p)}$ is found numerically to approximate ${\displaystyle 0.065770}$...

Show the effect of varying ${\displaystyle n}$ on the accuracy of simulated ${\displaystyle K(p)}$ for ${\displaystyle p=0.5}$ and for values of ${\displaystyle n}$ up to at least ${\displaystyle 1000}$. Any calculation of ${\displaystyle C_{n}}$ for finite ${\displaystyle n}$ is subject to randomness, so an approximation should be computed as the average of ${\displaystyle t}$ runs, where ${\displaystyle t}$${\displaystyle 5}$.

For extra credit, graphically show clusters in a ${\displaystyle 15\times 15}$, ${\displaystyle p=0.5}$ grid.

## 11l

Translation of: Nim
UInt32 seed = 0
F nonrandom()
:seed = 1664525 * :seed + 1013904223
R (:seed >> 16) / Float(FF'FF)

V nn = 15
V tt = 5
V pp = 0.5
V NotClustered = 1
V Cell2Char = ‘ #abcdefghijklmnopqrstuvwxyz’
V NRange = [4, 64, 256, 1024, 4096]

F newGrid(n, p)
R (0 .< n).map(i -> (0 .< @n).map(i -> Int(nonrandom() < @@p)))

F walkMaze(&grid, m, n, idx) -> Void
grid[n][m] = idx
I n < grid.len - 1 & grid[n + 1][m] == NotClustered
walkMaze(&grid, m, n + 1, idx)
I m < grid[0].len - 1 & grid[n][m + 1] == NotClustered
walkMaze(&grid, m + 1, n, idx)
I m > 0 & grid[n][m - 1] == NotClustered
walkMaze(&grid, m - 1, n, idx)
I n > 0 & grid[n - 1][m] == NotClustered
walkMaze(&grid, m, n - 1, idx)

F clusterCount(&grid)
V walkIndex = 1
L(n) 0 .< grid.len
L(m) 0 .< grid[0].len
I grid[n][m] == NotClustered
walkIndex++
walkMaze(&grid, m, n, walkIndex)
R walkIndex - 1

F clusterDensity(n, p)
V grid = newGrid(n, p)
R clusterCount(&grid) / Float(n * n)

F print_grid(grid)
L(row) grid
print(L.index % 10, end' ‘) ’)
L(cell) row
print(‘ ’Cell2Char[cell], end' ‘’)
print()

V grid = newGrid(nn, 0.5)
print(‘Found ’clusterCount(&grid)‘ clusters in this ’nn‘ by ’nn" grid\n")
print_grid(grid)
print()

L(n) NRange
V sum = 0.0
L 0 .< tt
sum += clusterDensity(n, pp)
V sim = sum / tt
print(‘t = #.  p = #.2  n = #4  sim = #.5’.format(tt, pp, n, sim))
Output:
Found 25 clusters in this 15 by 15 grid

0)  a a     b   c               d
1)      e e       d d d d     d d
2)  e e e e       d     d d     d
3)  e e e e   e e   e e   d d d d
4)  e   e e e e e   e e     d d d
5)      e       e e e e   f   d
6)  g   e     h   e     i     d
7)  g     j         k     k   d d
8)      l   m     k k k k k
9)  n   l   m   o   k k k k     p
0)  n             k k k k k   q
1)  n   r r     s       k   t   u
2)        r           k k k     u
3)  v   r r       w   k k k   x
4)  v   r r     w w w           y

t = 5  p = 0.50  n =    4  sim = 0.17500
t = 5  p = 0.50  n =   64  sim = 0.07300
t = 5  p = 0.50  n =  256  sim = 0.06823
t = 5  p = 0.50  n = 1024  sim = 0.06618
t = 5  p = 0.50  n = 4096  sim = 0.06590


## C

#include <stdio.h>
#include <stdlib.h>

int *map, w, ww;

void make_map(double p)
{
int i, thresh = RAND_MAX * p;
i = ww = w * w;

map = realloc(map, i * sizeof(int));
while (i--) map[i] = -(rand() < thresh);
}

char alpha[] = "+.ABCDEFGHIJKLMNOPQRSTUVWXYZ"
"abcdefghijklmnopqrstuvwxyz";
#define ALEN ((int)(sizeof(alpha) - 3))

void show_cluster(void)
{
int i, j, *s = map;

for (i = 0; i < w; i++) {
for (j = 0; j < w; j++, s++)
printf(" %c", *s < ALEN ? alpha[1 + *s] : '?');
putchar('\n');
}
}

void recur(int x, int v) {
if (x >= 0 && x < ww && map[x] == -1) {
map[x] = v;
recur(x - w, v);
recur(x - 1, v);
recur(x + 1, v);
recur(x + w, v);
}
}

int count_clusters(void)
{
int i, cls;

for (cls = i = 0; i < ww; i++) {
if (-1 != map[i]) continue;
recur(i, ++cls);
}

return cls;
}

double tests(int n, double p)
{
int i;
double k;

for (k = i = 0; i < n; i++) {
make_map(p);
k += (double)count_clusters() / ww;
}
return k / n;
}

int main(void)
{
w = 15;
make_map(.5);
printf("width=15, p=0.5, %d clusters:\n", count_clusters());
show_cluster();

printf("\np=0.5, iter=5:\n");
for (w = 1<<2; w <= 1<<14; w<<=2)
printf("%5d %9.6f\n", w, tests(5, .5));

free(map);
return 0;
}

Output:
width=15, p=0.5, 23 clusters:
A . . . B . C C C C . D . E .
A . . B B . . . . . . . . . .
A . . . . . F . . G . H . . I
. . J J J . . K K . L . M . I
. J J . . . K K K K . M M . .
. . . . K K . K . K . M . N .
O O . K K K . K . . . . N N N
. O O . K K K K K . P . N . .
Q . . K K . . . K . P . . . .
. R . K K . . K K . P . . S .
. . K K . . . . K . P . . . K
K K K K K . . K K . . T . . K
K . K . . . U . K . . T . . .
K . K K K K . K K K . T . . .
. . K . K . V . K K . . . W .

p=0.5, iter=5:
4  0.125000
16  0.083594
64  0.064453
256  0.066864
1024  0.065922
4096  0.065836
16384  0.065774


## C++

#include <iostream>
#include <random>
#include <string>
#include <vector>
#include <iomanip>

std::random_device random;
std::mt19937 generator(random());
std::uniform_real_distribution<double> distribution(0.0F, 1.0F);

class Grid {
public:
Grid(const int32_t size, const double probability) {
create_grid(size, probability);
count_clusters();
}

int32_t cluster_count() const {
return clusters;
}

double cluster_density() const {
return (double) clusters / ( grid.size() * grid.size() );
}

void display() const {
for ( uint64_t row = 0; row < grid.size(); ++row ) {
for ( uint64_t col = 0; col < grid.size(); ++col ) {
uint64_t value = grid[row][col];
char ch = ( value < GRID_CHARACTERS.length() ) ? GRID_CHARACTERS[value] : '?';
std::cout << " " << ch;
}
std::cout << std::endl;
}
}
private:
void count_clusters() {
clusters = 0;
for ( uint64_t row = 0; row < grid.size(); ++row ) {
for ( uint64_t col = 0; col < grid.size(); ++col ) {
if ( grid[row][col] == CLUSTERED ) {
clusters += 1;
identify_cluster(row, col, clusters);
}
}
}
}

void identify_cluster(const uint64_t row, const uint64_t col, const uint64_t count) {
grid[row][col] = count;
if ( row < grid.size() - 1 && grid[row + 1][col] == CLUSTERED ) {
identify_cluster(row + 1, col, count);
}
if ( col < grid.size() - 1 && grid[row][col + 1] == CLUSTERED ) {
identify_cluster(row, col + 1, count);
}
if ( col > 0 && grid[row][col - 1] == CLUSTERED ) {
identify_cluster(row, col - 1, count);
}
if ( row > 0 && grid[row - 1][col] == CLUSTERED ) {
identify_cluster(row - 1, col, count);
}
}

void create_grid(int32_t grid_size, double probability) {
grid.assign(grid_size, std::vector<int32_t>(grid_size, 0));
for ( int32_t row = 0; row < grid_size; ++row ) {
for ( int32_t col = 0; col < grid_size; ++col ) {
if ( distribution(generator) < probability ) {
grid[row][col] = CLUSTERED;
}
}
}
}

int32_t clusters;
std::vector<std::vector<int32_t>> grid;

inline static const int CLUSTERED = -1;
inline static const std::string GRID_CHARACTERS = ".ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz";
};

int main() {
const int32_t size = 15;
const double probability = 0.5;
const int32_t test_count = 5;

Grid grid(size, probability);
std::cout << "This " << size << " by " << size << " grid contains "
<< grid.cluster_count() << " clusters:" << std::endl;
grid.display();

std::cout << "\n p = 0.5, iterations = " << test_count << std::endl;
std::vector<int32_t> grid_sizes = { 10, 100, 1'000, 10'000 };
for ( int32_t grid_size : grid_sizes ) {
double sumDensity = 0.0;
for ( int32_t test = 0; test < test_count; test++ ) {
Grid grid(grid_size, probability);
sumDensity += grid.cluster_density();
}
double result = sumDensity / test_count;
std::cout << " n = " << std::setw(5) << grid_size
<< ", simulations K = " << std::fixed << result << std::endl;
}
}

Output:
This 15 by 15 grid contains 11 clusters:
A A . B B . B B . . . . . . .
. . B B B B . B B . B B . . .
. C . . B B . B B B B . . . .
. C C . B . B B B B B B . D D
C C . . B B B B . . B . D D .
C C . . B B B . B B B . . D D
C C . . . . B . . . B . D D D
. C . E . . . . D D . . D D .
F . . . G . H . D D D D D D D
F F . F . I . F . D D . D D .
F . F F F . . F . . D D . . D
F . F . F . . F . F . D D D D
F F . . F F F F F F . D . . D
F F F F F . . F . . . D . . D
. F F . . J J . . . . D . K .

p = 0.5, iterations = 5
n =    10, simulation K = 0.088000
n =   100, simulation K = 0.067260
n =  1000, simulation K = 0.066215
n = 10000, simulation K = 0.065777


## D

Translation of: python
import std.stdio, std.algorithm, std.random, std.math, std.array,
std.range, std.ascii;

alias Cell = ubyte;
alias Grid = Cell[][];
enum Cell notClustered = 1; // Filled cell, but not in a cluster.

Grid initialize(Grid grid, in double prob, ref Xorshift rng) nothrow {
foreach (row; grid)
foreach (ref cell; row)
cell = Cell(rng.uniform01 < prob);
return grid;
}

void show(in Grid grid) {
immutable static cell2char = " #" ~ letters;
writeln('+', "-".replicate(grid.length), '+');
foreach (row; grid) {
write('|');
row.map!(c => c < cell2char.length ? cell2char[c] : '@').write;
writeln('|');
}
writeln('+', "-".replicate(grid.length), '+');
}

size_t countClusters(bool justCount=false)(Grid grid)
pure nothrow @safe @nogc {
immutable side = grid.length;
static if (justCount)
enum Cell clusterID = 2;
else
Cell clusterID = 1;

void walk(in size_t r, in size_t c) nothrow @safe @nogc {
grid[r][c] = clusterID; // Fill grid.

if (r < side - 1 && grid[r + 1][c] == notClustered) // Down.
walk(r + 1, c);
if (c < side - 1 && grid[r][c + 1] == notClustered) // Right.
walk(r, c + 1);
if (c > 0 && grid[r][c - 1] == notClustered) // Left.
walk(r, c - 1);
if (r > 0 && grid[r - 1][c] == notClustered) // Up.
walk(r - 1, c);
}

size_t nClusters = 0;

foreach (immutable r; 0 .. side)
foreach (immutable c; 0 .. side)
if (grid[r][c] == notClustered) {
static if (!justCount)
clusterID++;
nClusters++;
walk(r, c);
}
return nClusters;
}

double clusterDensity(Grid grid, in double prob, ref Xorshift rng) {
return grid.initialize(prob, rng).countClusters!true /
double(grid.length ^^ 2);
}

void showDemo(in size_t side, in double prob, ref Xorshift rng) {
auto grid = new Grid(side, side);
grid.initialize(prob, rng);
writefln("Found %d clusters in this %d by %d grid:\n",
grid.countClusters, side, side);
grid.show;
}

void main() {
immutable prob = 0.5;
immutable nIters = 5;
auto rng = Xorshift(unpredictableSeed);

showDemo(15, prob, rng);
writeln;
foreach (immutable i; iota(4, 14, 2)) {
immutable side = 2 ^^ i;
auto grid = new Grid(side, side);
immutable density = nIters
.iota
.map!(_ => grid.clusterDensity(prob, rng))
.sum / nIters;
writefln("n_iters=%3d, p=%4.2f, n=%5d, sim=%7.8f",
nIters, prob, side, density);
}
}

Output:
Found 26 clusters in this 15 by 15 grid:

+---------------+
| AA B    CCCC  |
|AA D E F CC  G |
|  DDD FF  CC  H|
| I D FF  J   K |
|  L  FF JJJJ   |
|L LLL      J  M|
|LLLLLL   JJJ MM|
|L LL L N  J   M|
|LL    O P J   M|
|LLL QQ R JJ  S |
|LL T  RR  J SSS|
| L   U  V JJ  S|
|  WW  XX JJ YY |
|    XXX   JJ YY|
|ZZ   XXX   JJ  |
+---------------+

n_iters=  5, p=0.50, n=   16, sim=0.09765625
n_iters=  5, p=0.50, n=   64, sim=0.07260742
n_iters=  5, p=0.50, n=  256, sim=0.06679993
n_iters=  5, p=0.50, n= 1024, sim=0.06609497
n_iters=  5, p=0.50, n= 4096, sim=0.06580237

Increasing the index i to 15:

n_iters=  5, p=0.50, n=32768, sim=0.06578374

## EchoLisp

We use the canvas bit-map as 2D-matrix. For extra-extra credit, a 800x800 nice cluster tapestry image is shown here : http://www.echolalie.org/echolisp/images/rosetta-clusters-800.png.

(define-constant BLACK  (rgb 0 0 0.6))
(define-constant WHITE -1)
;; sets pixels to clusterize to WHITE
;; returns bit-map vector
(define (init-C n p )
(plot-size n n)
(define C (pixels->int32-vector )) ;; get canvas bit-map
(pixels-map (lambda (x y) (if (< (random) p) WHITE BLACK )) C)
C )

;; random color for new cluster
(define (new-color)
(hsv->rgb (random) 0.9 0.9))

;; make-region predicate
(define (in-cluster C x y)
(= (pixel-ref C x y) WHITE))

;; paint all adjacents to (x0,y0) with new color
(define (make-cluster C x0 y0)
(pixel-set! C x0 y0 (new-color))
(make-region in-cluster C x0 y0))

(define (make-clusters (n 400) (p 0.5))
(define Cn 0)
(define C null)
(for ((t 5)) ;; 5 iterations
(plot-clear)
(set!  C (init-C n p))
(for* ((x0 n) (y0 n))
#:when  (= (pixel-ref C x0 y0) WHITE)
(set! Cn (1+ Cn))
(make-cluster C x0 y0)))

(writeln 'n n  'Cn Cn  'density  (// Cn (* n n) 5) )
(vector->pixels C)) ;; to screen

Output:
n     100      Cn     3420       density     0.0684
n     400      Cn     53246      density     0.0665575
n     600      Cn     118346     density     0.06574778
n     800      Cn     212081     density     0.0662753125
n     1000     Cn     330732     density     0.0661464


## Factor

USING: combinators formatting generalizations kernel math
math.matrices random sequences ;
IN: rosetta-code.mean-cluster-density

CONSTANT: p 0.5
CONSTANT: iterations 5

: rand-bit-matrix ( n probability -- matrix )
dupd [ random-unit > 1 0 ? ] curry make-matrix ;

: flood-fill ( x y matrix -- )
3dup ?nth ?nth 1 = [
[ [ -1 ] 3dip nth set-nth ] [
{
[ [ 1 + ] 2dip ]
[ [ 1 - ] 2dip ]
[ [ 1 + ] dip ]
[ [ 1 - ] dip ]
} [ flood-fill ] map-compose 3cleave
] 3bi
] [ 3drop ] if ;

: count-clusters ( matrix -- Cn )
0 swap dup dim matrix-coordinates flip concat [
first2 rot 3dup ?nth ?nth 1 = [ flood-fill 1 + ]
[ 3drop ] if
] with each ;

: mean-cluster-density ( matrix -- mcd )
[ count-clusters ] [ dim first sq / ] bi ;

: simulate ( n -- avg-mcd )
iterations swap [ p rand-bit-matrix mean-cluster-density ]
curry replicate sum iterations / ;

: main ( -- )
{ 4 64 256 1024 4096 } [
[ iterations p ] dip dup simulate
"iterations = %d p = %.1f n = %4d sim = %.5f\n" printf
] each ;

MAIN: main

Output:
iterations = 5 p = 0.5 n =    4 sim = 0.13750
iterations = 5 p = 0.5 n =   64 sim = 0.07437
iterations = 5 p = 0.5 n =  256 sim = 0.06786
iterations = 5 p = 0.5 n = 1024 sim = 0.06621
iterations = 5 p = 0.5 n = 4096 sim = 0.06589


## Go

Translation of: Python
package main

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

var (
n_range = []int{4, 64, 256, 1024, 4096}
M       = 15
N       = 15
)

const (
p             = .5
t             = 5
NOT_CLUSTERED = 1
cell2char     = " #abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ"
)

func newgrid(n int, p float64) [][]int {
g := make([][]int, n)
for y := range g {
gy := make([]int, n)
for x := range gy {
if rand.Float64() < p {
gy[x] = 1
}
}
g[y] = gy
}
return g
}

func pgrid(cell [][]int) {
for n := 0; n < N; n++ {
fmt.Print(n%10, ") ")
for m := 0; m < M; m++ {
fmt.Printf(" %c", cell2char[cell[n][m]])
}
fmt.Println()
}
}

func cluster_density(n int, p float64) float64 {
cc := clustercount(newgrid(n, p))
return float64(cc) / float64(n) / float64(n)
}

func clustercount(cell [][]int) int {
walk_index := 1
for n := 0; n < N; n++ {
for m := 0; m < M; m++ {
if cell[n][m] == NOT_CLUSTERED {
walk_index++
walk_maze(m, n, cell, walk_index)
}
}
}
return walk_index - 1
}

func walk_maze(m, n int, cell [][]int, indx int) {
cell[n][m] = indx
if n < N-1 && cell[n+1][m] == NOT_CLUSTERED {
walk_maze(m, n+1, cell, indx)
}
if m < M-1 && cell[n][m+1] == NOT_CLUSTERED {
walk_maze(m+1, n, cell, indx)
}
if m > 0 && cell[n][m-1] == NOT_CLUSTERED {
walk_maze(m-1, n, cell, indx)
}
if n > 0 && cell[n-1][m] == NOT_CLUSTERED {
walk_maze(m, n-1, cell, indx)
}
}

func main() {
rand.Seed(time.Now().Unix())
cell := newgrid(N, .5)
fmt.Printf("Found %d clusters in this %d by %d grid\n\n",
clustercount(cell), N, N)
pgrid(cell)
fmt.Println()

for _, n := range n_range {
M = n
N = n
sum := 0.
for i := 0; i < t; i++ {
sum += cluster_density(n, p)
}
sim := sum / float64(t)
fmt.Printf("t=%3d p=%4.2f n=%5d sim=%7.5f\n", t, p, n, sim)
}
}

Output:
Found 29 clusters in this 15 by 15 grid

0)          a a a   b     c
1)  d     e     a       c c     f
2)    g   e e e
3)      h   e       i     j     k
4)  l   h             m m     n
5)  l           o   p     n n n n
6)      q     o o o   r   n
7)          o o     r r       s
8)  t t     o   u     r   s s s s
9)  t t   v   u u   r r   s s s s
0)    t   v   u     r   r   s
1)      w       x   r   r     y y
2)          z   x   r r r r r   y
3)    A   z z     B     r r r r
4)  A A A   z   C       r   r r r

t=  5 p=0.50 n=    4 sim=0.16250
t=  5 p=0.50 n=   64 sim=0.07334
t=  5 p=0.50 n=  256 sim=0.06710
t=  5 p=0.50 n= 1024 sim=0.06619
t=  5 p=0.50 n= 4096 sim=0.06585


{-# language FlexibleContexts #-}
import Data.List
import Data.Maybe
import System.Random
import Text.Printf
import Data.Set (Set)
import qualified Data.Set  as S

type Matrix = [[Bool]]
type Cell = (Int, Int)
type Cluster = Set (Int, Int)

clusters :: Matrix -> [Cluster]
clusters m = unfoldr findCuster cells
where
cells = S.fromList [ (i,j) | (r, i) <- zip m [0..]
, (x, j) <- zip r [0..], x]

findCuster s = do
(p, ps) <- S.minView s
return $runState (expand p) ps expand p = do ns <- state$ extract (neigbours p)
xs <- mapM expand $S.elems ns return$ S.insert p $mconcat xs extract s1 s2 = (s2 S.intersection s1, s2 S.\\ s1) neigbours (i,j) = S.fromList [(i-1,j),(i+1,j),(i,j-1),(i,j+1)] n = length m showClusters :: Matrix -> String showClusters m = unlines [ unwords [ mark (i,j) | j <- [0..n-1] ] | i <- [0..n-1] ] where cls = clusters m n = length m mark c = maybe "." snd$ find (S.member c . fst) $zip cls syms syms = sequence [['a'..'z'] ++ ['A'..'Z']] ------------------------------------------------------------ randomMatrices :: Int -> StdGen -> [Matrix] randomMatrices n = clipBy n . clipBy n . randoms where clipBy n = unfoldr (Just . splitAt n) randomMatrix n = head . randomMatrices n tests :: Int -> StdGen -> [Int] tests n = map (length . clusters) . randomMatrices n task :: Int -> StdGen -> (Int, Double) task n g = (n, result) where result = mean$ take 10 $map density$ tests n g
density c = fromIntegral c / fromIntegral n**2
mean lst = sum lst / genericLength lst

main = newStdGen >>= mapM_ (uncurry (printf "%d\t%.5f\n")) . res
where

λ> newStdGen >>= putStrLn . showClusters . randomMatrix 15
. . a a . b b b b . . c c . .
d d . . . . . . b b . c . . .
d . . e . . . b b . c c . f f
d d d . g g . b b . c c c . .
. d . d . . b b . h . . c . i
d d . d d d . . h h h h . . i
d d d d . d . . h . . . . i i
. . . d . d . . h . i i i i i
. j . d . . . . . k . i . i .
. . l . . . . k k k k . . i i
m m . m . . . k k . . n . i .
m m m m m . o . k . n n . . .
. m . m . p . k k . . n . . q
. m m . . . r . k . . n n . q
. m . s s . r r . t . . . . q

λ> take 10 $tests 15 (mkStdGen 42) [33,18,26,18,29,14,23,21,18,24] λ> main 10 0.10100 50 0.07072 100 0.06878 500 0.06676 ## J The first thing this task seems to need is some mechanism of identifying "clusters", using "percolation". We can achieve this by assigning every "1" in a matrix a unique integer value and then defining an operation which combines two numbers - doing nothing unless the second one (the one on the right) is non-zero. If it is non-zero we pick the larger of the two values. (*@[ * >.) Once we have this, we can identify clusters by propagating information in a single direction through the matrix using this operation, rotating the matrix 90 degrees, and then repeating this combination of operations four times. And, finally, by keeping at this until there's nothing more to be done. congeal=: |.@|:@((*@[*>.)/\.)^:4^:_  Example:  M=:0.4>?6 6$0
M
1 0 0 0 0 0
0 0 0 1 0 0
0 0 0 1 0 0
1 1 0 0 0 0
0 0 0 1 0 1
1 1 0 1 1 0
M*p:i.$M 2 0 0 0 0 0 0 0 0 29 0 0 0 0 0 53 0 0 67 71 0 0 0 0 0 0 0 107 0 113 127 131 0 139 149 0 congeal M*p:i.$M
2   0 0   0   0   0
0   0 0  53   0   0
0   0 0  53   0   0
71  71 0   0   0   0
0   0 0 149   0 113
131 131 0 149 149   0


We did not have to use primes there - any mechanism for assigning distinct positive integers to the 1s would work. And, in fact, it might be nice if - once we found our clusters - we assigned the smallest distinct positive integers to the clusters. This would allow us to use simple indexing to map the array to characters.

idclust=:  [: (~. i.])&.(0&,)@,@congeal  ] * 1 + i.@$ Example use:  idclust M 1 0 0 0 0 0 0 0 0 2 0 0 0 0 0 2 0 0 3 3 0 0 0 0 0 0 0 4 0 5 6 6 0 4 4 0 (idclust M) {'.ABCDEFG' A..... ...B.. ...B.. CC.... ...D.E FF.DD.  Now we just need a measure of cluster density. Formally cluster density seems to be defined as the number of clusters divided by the total number of elements of the matrix. Thus: K=: (%&#~ }.@~.)&,  Example use:  K idclust M 0.1666667  So we can create a word that performs a simulation experiment, given a probability getting a 1 and the number of rows (and columns) of our square matrix M. experiment=: K@ idclust@: > 0 ?@$~ ,~


Example use:

   0.4 experiment 6
0.1666667
0.4 experiment 6
0.1944444


The task wants us to perform at least five trials for sizes up to 1000 by 1000 with probability of 1 being 0.5:

trials=: 0.5&experiment"0@#


Example use:

   6 trials 3
0.1111111 0.1111111 0.2222222 0.1111111 0.1111111 0.3333333
6 trials 10
0.16 0.12 0.09 0.1 0.1 0.03
6 trials 30
0.05666667 0.1033333 0.08222222 0.07444444 0.08333333 0.07666667
6 trials 100
0.069 0.0678 0.0666 0.0677 0.0653 0.0739
6 trials 300
0.06563333 0.06663333 0.06713333 0.06727778 0.06658889 0.06664444
6 trials 1000
0.066079 0.066492 0.065847 0.065943 0.066318 0.065998


Now for averages (these are different trials from the above):

mean=: +/%#
mean 8 trials 3
0.1805556
mean 8 trials 10
0.0875
mean 8 trials 30
0.07486111
mean 8 trials 100
0.0690625
mean 8 trials 300
0.06749861
mean 8 trials 1000
0.06616738


Finally, for the extra credit (thru taken from the Loops/Downward for task):

thru=: <./ + i.@(+*)@-~

   (idclust 0.5 > 0 ?@$~ 15 15) {'.', 'A' thru&.(a.&i.) 'Z' A.......B..C... AAAA...D..E.F.. A..A.G.D.D.FFF. AA..H..DDD.FF.I AAA...J...FFF.. ..AAAA.A.K...AA LL.A...A..A.AAA .L.A..AAA.AAAAA ..AA.AAA.AAA.A. AA.AAAAAA....A. A.AAAA.AAAA.AA. AAA...AAA.AAAAA ..AA..A.A...AAA .M.A.AA.AA..AA. .MM..A.N..O..A.  Collected definitions congeal=: |.@|:@((*@[*>.)/\.)^:4^:_ idclust=:$ $[: (~. i.])&.(0&,)@,@congeal ] * 1 + i.@$

K=: (%&#~ }.@~.)&,

experiment=: K@ idclust@: > 0 ?@$~ ,~ trials=: 0.5&experiment"0@# mean=:+/ % # thru=: <./ + i.@(+*)@-~  Extra Credit  M=: (* 1+i.@$)?15 15$2 M 0 2 3 4 0 6 0 8 0 10 11 12 0 0 15 0 0 18 19 20 0 22 0 0 0 0 0 28 29 0 31 32 0 34 35 36 37 38 0 0 0 42 0 0 45 0 0 48 49 0 51 0 0 54 55 0 57 58 0 0 61 62 63 64 0 0 67 0 69 0 71 72 0 74 0 0 0 78 79 0 0 82 0 84 85 86 87 88 0 0 0 92 0 94 0 0 0 0 99 100 101 0 103 0 105 106 107 108 0 0 111 0 0 114 115 116 0 0 0 0 0 0 0 124 125 126 127 0 0 0 0 0 133 134 135 0 0 138 0 0 141 0 143 144 145 0 0 0 0 150 0 152 153 154 0 0 0 158 0 160 0 162 163 164 165 0 167 168 169 170 0 172 173 0 175 176 177 0 0 180 181 182 183 0 0 186 0 188 189 190 191 192 0 194 195 196 197 198 0 200 201 202 0 0 205 0 207 0 0 0 211 212 213 0 0 0 217 218 0 220 221 0 0 224 0 congeal M 0 94 94 94 0 6 0 8 0 12 12 12 0 0 15 0 0 94 94 94 0 94 0 0 0 0 0 29 29 0 32 32 0 94 94 94 94 94 0 0 0 116 0 0 45 0 0 94 94 0 94 0 0 116 116 0 116 116 0 0 94 94 94 94 0 0 82 0 116 0 116 116 0 74 0 0 0 94 94 0 0 82 0 116 116 116 116 116 0 0 0 108 0 94 0 0 0 0 116 116 116 0 116 0 105 108 108 108 0 0 141 0 0 116 116 116 0 0 0 0 0 0 0 141 141 141 141 0 0 0 0 0 221 221 221 0 0 213 0 0 141 0 221 221 221 0 0 0 0 221 0 213 213 213 0 0 0 221 0 221 0 221 221 221 221 0 213 213 213 213 0 221 221 0 221 221 221 0 0 221 213 213 213 0 0 218 0 221 221 221 221 221 0 221 221 213 213 213 0 218 218 218 0 0 221 0 221 0 0 0 213 213 213 0 0 0 218 218 0 221 221 0 0 224 0 (~.@, i. ])congeal M 0 1 1 1 0 2 0 3 0 4 4 4 0 0 5 0 0 1 1 1 0 1 0 0 0 0 0 6 6 0 7 7 0 1 1 1 1 1 0 0 0 8 0 0 9 0 0 1 1 0 1 0 0 8 8 0 8 8 0 0 1 1 1 1 0 0 10 0 8 0 8 8 0 11 0 0 0 1 1 0 0 10 0 8 8 8 8 8 0 0 0 12 0 1 0 0 0 0 8 8 8 0 8 0 13 12 12 12 0 0 14 0 0 8 8 8 0 0 0 0 0 0 0 14 14 14 14 0 0 0 0 0 15 15 15 0 0 16 0 0 14 0 15 15 15 0 0 0 0 15 0 16 16 16 0 0 0 15 0 15 0 15 15 15 15 0 16 16 16 16 0 15 15 0 15 15 15 0 0 15 16 16 16 0 0 17 0 15 15 15 15 15 0 15 15 16 16 16 0 17 17 17 0 0 15 0 15 0 0 0 16 16 16 0 0 0 17 17 0 15 15 0 0 18 0  ## Java import java.util.List; import java.util.concurrent.ThreadLocalRandom; public final class PercolationMeanCluster { public static void main(String[] aArgs) { final int size = 15; final double probability = 0.5; final int testCount = 5; Grid grid = new Grid(size, probability); System.out.println("This " + size + " by " + size + " grid contains " + grid.clusterCount() + " clusters:"); grid.display(); System.out.println(System.lineSeparator() + " p = 0.5, iterations = " + testCount); List<Integer> gridSizes = List.of( 10, 100, 1_000, 10_000 ); for ( int gridSize : gridSizes ) { double sumDensity = 0.0; for ( int test = 0; test < testCount; test++ ) { grid = new Grid(gridSize, probability); sumDensity += grid.clusterDensity(); } double result = sumDensity / testCount; System.out.println(String.format("%s%5d%s%.6f", " n = ", gridSize, ", simulation K = ", result)); } } } final class Grid { public Grid(int aSize, double aProbability) { createGrid(aSize, aProbability); countClusters(); } public int clusterCount() { return clusterCount; } public double clusterDensity() { return (double) clusterCount / ( grid.length * grid.length ); } public void display() { for ( int row = 0; row < grid.length; row++ ) { for ( int col = 0; col < grid.length; col++ ) { int value = grid[row][col]; char ch = ( value < GRID_CHARACTERS.length() ) ? GRID_CHARACTERS.charAt(value) : '?'; System.out.print(" " + ch); } System.out.println(); } } private void countClusters() { clusterCount = 0; for ( int row = 0; row < grid.length; row++ ) { for ( int col = 0; col < grid.length; col++ ) { if ( grid[row][col] == CLUSTERED ) { clusterCount += 1; identifyCluster(row, col, clusterCount); } } } } private void identifyCluster(int aRow, int aCol, int aCount) { grid[aRow][aCol] = aCount; if ( aRow < grid.length - 1 && grid[aRow + 1][aCol] == CLUSTERED ) { identifyCluster(aRow + 1, aCol, aCount); } if ( aCol < grid[0].length - 1 && grid[aRow][aCol + 1] == CLUSTERED ) { identifyCluster(aRow, aCol + 1, aCount); } if ( aCol > 0 && grid[aRow][aCol - 1] == CLUSTERED ) { identifyCluster(aRow, aCol - 1, aCount); } if ( aRow > 0 && grid[aRow - 1][aCol] == CLUSTERED ) { identifyCluster(aRow - 1, aCol, aCount); } } private void createGrid(int aGridSize, double aProbability) { grid = new int[aGridSize][aGridSize]; for ( int row = 0; row < aGridSize; row++ ) { for ( int col = 0; col < aGridSize; col++ ) { if ( random.nextDouble(1.0) < aProbability ) { grid[row][col] = CLUSTERED; } } } } private int[][] grid; private int clusterCount; private static ThreadLocalRandom random = ThreadLocalRandom.current(); private static final int CLUSTERED = -1; private static final String GRID_CHARACTERS = ".ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz"; }  Output: This 15 by 15 grid contains 21 clusters: A . . B B B . . C . D D . E . . . F . B . G . C . . . E E E . F F . B . . . C C . H . E . F F . B B B . I . C C . . E . . . . . . . . I . . . J . . K . L . . M M . I . . N . . . K O . . O . M . I I . . . . K K O O O O O . I I . K K . . . K . O . O . P . I . K . K K K K . . Q . . . I I . K K K K K K Q Q Q . . I I I I . . . . . K Q . Q Q . . I . . . . R R . . . . Q . I I I . . . . R . R R . . Q . . I I . . . . R R R . S S . T . . I I I . R R R . U p = 0.5, iterations = 5 n = 10, simulation K = 0.094000 n = 100, simulation K = 0.070420 n = 1000, simulation K = 0.066056 n = 10000, simulation K = 0.065780  ## Julia Translation of: Python using Printf, Distributions newgrid(p::Float64, r::Int, c::Int=r) = rand(Bernoulli(p), r, c) function walkmaze!(grid::Matrix{Int}, r::Int, c::Int, indx::Int) NOT_CLUSTERED = 1 # const N, M = size(grid) dirs = [[1, 0], [-1, 0], [0, 1], [0, -1]] # fill cell grid[r, c] = indx # check for each direction for d in dirs rr, cc = (r, c) .+ d if checkbounds(Bool, grid, rr, cc) && grid[rr, cc] == NOT_CLUSTERED walkmaze!(grid, rr, cc, indx) end end end function clustercount!(grid::Matrix{Int}) NOT_CLUSTERED = 1 # const walkind = 1 for r in 1:size(grid, 1), c in 1:size(grid, 2) if grid[r, c] == NOT_CLUSTERED walkind += 1 walkmaze!(grid, r, c, walkind) end end return walkind - 1 end clusterdensity(p::Float64, n::Int) = clustercount!(newgrid(p, n)) / n ^ 2 function printgrid(G::Matrix{Int}) LETTERS = vcat(' ', '#', 'A':'Z', 'a':'z') for r in 1:size(G, 1) println(r % 10, ") ", join(LETTERS[G[r, :] .+ 1], ' ')) end end G = newgrid(0.5, 15) @printf("Found %i clusters in this %i×%i grid\n\n", clustercount!(G), size(G, 1), size(G, 2)) printgrid(G) println() const nrange = 2 .^ (4:2:12) const p = 0.5 const nrep = 5 for n in nrange sim = mean(clusterdensity(p, n) for _ in 1:nrep) @printf("nrep = %2i p = %.2f dim = %-13s sim = %.5f\n", nrep, p, "$n × $n", sim) end  Output: Found 20 clusters in this 15×15 grid 1) A B C C D D D 2) E F D D 3) G F D D D D D D 4) G G H F I D D D J 5) G G K L D J 6) G G G G M N N 7) G G G G G G O O O N 8) G G O N N N 9) P P P G G G N N 0) P P P P G Q Q Q N 1) P Q Q Q Q N N 2) P N N N N 3) P P P R S N 4) P R R R S S N N 5) R R R S T N nrep = 5 p = 0.50 dim = 16 × 16 sim = 0.07500 nrep = 5 p = 0.50 dim = 64 × 64 sim = 0.07178 nrep = 5 p = 0.50 dim = 256 × 256 sim = 0.06690 nrep = 5 p = 0.50 dim = 1024 × 1024 sim = 0.06609 nrep = 5 p = 0.50 dim = 4096 × 4096 sim = 0.06588 ## Kotlin Translation of: C // version 1.2.10 import java.util.Random val rand = Random() const val RAND_MAX = 32767 lateinit var map: IntArray var w = 0 var ww = 0 const val ALPHA = "+.ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz" const val ALEN = ALPHA.length - 3 fun makeMap(p: Double) { val thresh = (p * RAND_MAX).toInt() ww = w * w var i = ww map = IntArray(i) while (i-- != 0) { val r = rand.nextInt(RAND_MAX + 1) if (r < thresh) map[i] = -1 } } fun showCluster() { var k = 0 for (i in 0 until w) { for (j in 0 until w) { val s = map[k++] val c = if (s < ALEN) ALPHA[1 + s] else '?' print("$c")
}
println()
}
}

fun recur(x: Int, v: Int) {
if ((x in 0 until ww) && map[x] == -1) {
map[x] = v
recur(x - w, v)
recur(x - 1, v)
recur(x + 1, v)
recur(x + w, v)
}
}

fun countClusters(): Int {
var cls = 0
for (i in 0 until ww) {
if (map[i] != -1) continue
recur(i, ++cls)
}
return cls
}

fun tests(n: Int, p: Double): Double {
var k = 0.0
for (i in 0 until n) {
makeMap(p)
k += countClusters().toDouble() / ww
}
return k / n
}

fun main(args: Array<String>) {
w = 15
makeMap(0.5)
val cls = countClusters()
println("width = 15, p = 0.5, $cls clusters:") showCluster() println("\np = 0.5, iter = 5:") w = 1 shl 2 while (w <= 1 shl 13) { val t = tests(5, 0.5) println("%5d %9.6f".format(w, t)) w = w shl 1 } }  Sample output: width = 15, p = 0.5, 23 clusters: A . B . C C . . . . D D . D . . B B B . . D D D . D D . D D . B . B B B . . D D D . . D . . . E . . B . B . . D D D D . F . . B B B B B . G . D . D D . . B B . B . . H . D D . D D . B B . . . I . H H . . D D . B B B . . J . K . . L . D . B B . . M . . K K K . . D D . B B B . . . . . K K . . D . B B B . . N N . . . . . . . O . . B . . . . P . . O O O O O . . . . Q . . P . . . O O O O . . . . . R . . . . S . O . . T T . U . . . V . . . . . W . . . p = 0.5, iter = 5: 4 0.112500 8 0.121875 16 0.075000 32 0.068750 64 0.068164 128 0.065625 256 0.067093 512 0.065815 1024 0.065863 2048 0.065815 4096 0.065764 8192 0.065766  ## Mathematica /Wolfram Language (*Calculate C_n / n^2 for n=1000, 2000, ..., 10 000*) In[1]:= Table[N[Max@MorphologicalComponents[ RandomVariate[BernoulliDistribution[.5], {n, n}], CornerNeighbors -> False]/n^2], {n, 10^3, 10^4, 10^3}] (*Find the average*) In[2]:= % // MeanAround (*Show a 15x15 matrix with each cluster given an incrementally higher number, Colorize instead of MatrixForm creates an image*) In[3]:= MorphologicalComponents[RandomChoice[{0, 1}, {15, 15}], CornerNeighbors -> False] // MatrixForm  Output: Out[1]= {0.066339, 0.06568, 0.0656282, 0.0658778, 0.0657444, 0.0658455, 0.06578, 0.0657307, 0.0658186, 0.0657963} Out[2]= 0.06582 +- 0.00006 Out[3]= (1 1 0 2 2 2 2 2 0 0 2 2 2 2 0 0 0 3 0 0 0 2 2 0 0 0 2 2 2 0 3 3 3 3 3 0 0 2 2 2 2 2 2 0 0 0 0 0 0 0 4 4 0 2 0 0 2 0 0 0 0 7 7 0 5 0 0 0 2 0 0 2 0 0 6 7 7 0 0 0 7 7 0 0 8 0 2 2 0 6 0 7 7 0 7 7 7 0 8 8 8 0 0 9 0 10 0 7 7 7 7 7 7 0 8 0 11 0 9 9 0 0 7 0 7 0 0 7 7 0 11 11 0 0 0 0 0 0 7 7 7 7 7 0 0 11 0 12 12 0 0 13 0 7 0 7 7 0 0 14 0 14 0 0 16 15 0 7 7 0 7 7 7 0 14 14 14 0 16 16 0 0 0 7 7 7 7 0 17 0 14 14 14 0 0 0 18 0 7 7 0 0 0 0 0 0 0 0 19 0 0 0 0 0 7 0 0 20 0 0 21 21 0 0 22) ## Nim Translation of: Go import random, sequtils, strformat const N = 15 T = 5 P = 0.5 NotClustered = 1 Cell2Char = " #abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ" NRange = [4, 64, 256, 1024, 4096] type Grid = seq[seq[int]] proc newGrid(n: Positive; p: float): Grid = result = newSeqWith(n, newSeq[int](n)) for row in result.mitems: for cell in row.mitems: if rand(1.0) < p: cell = 1 func walkMaze(grid: var Grid; m, n, idx: int) = grid[n][m] = idx if n < grid.high and grid[n + 1][m] == NotClustered: grid.walkMaze(m, n + 1, idx) if m < grid[0].high and grid[n][m + 1] == NotClustered: grid.walkMaze(m + 1, n, idx) if m > 0 and grid[n][m - 1] == NotClustered: grid.walkMaze(m - 1, n, idx) if n > 0 and grid[n - 1][m] == NotClustered: grid.walkMaze(m, n - 1, idx) func clusterCount(grid: var Grid): int = var walkIndex = 1 for n in 0..grid.high: for m in 0..grid[0].high: if grid[n][m] == NotClustered: inc walkIndex grid.walkMaze(m, n, walkIndex) result = walkIndex - 1 proc clusterDensity(n: int; p: float): float = var grid = newGrid(n, p) result = grid.clusterCount() / (n * n) proc print(grid: Grid) = for n, row in grid: stdout.write n mod 10, ") " for cell in row: stdout.write ' ', Cell2Char[cell] stdout.write '\n' when isMainModule: randomize() var grid = newGrid(N, 0.5) echo &"Found {grid.clusterCount()} clusters in this {N} by {N} grid\n" grid.print() echo "" for n in NRange: var sum = 0.0 for _ in 1..T: sum += clusterDensity(n, P) let sim = sum / T echo &"t = {T} p = {P:4.2f} n = {n:4} sim = {sim:7.5f}"  Output: Found 14 clusters in this 15 by 15 grid 0) a a b c c c c d 1) e e c c c c c c c 2) f e c c c c c 3) f f c c c c c c c c c 4) f c c c c c c c c c c c 5) g c c c c c c c 6) h i c c c c c c c c 7) i i c c c 8) j j j j c c k 9) l j c k k 0) l l l l j j k 1) l l l l l l j j j j j 2) l l l m j j j j 3) l l l l j j 4) n l l l l j j j j j j j j t = 5 p = 0.50 n = 4 sim = 0.11250 t = 5 p = 0.50 n = 64 sim = 0.07085 t = 5 p = 0.50 n = 256 sim = 0.06702 t = 5 p = 0.50 n = 1024 sim = 0.06619 t = 5 p = 0.50 n = 4096 sim = 0.06587 ## Perl Translation of: Raku $fill = 'x';
$D{$_} = $i++ for qw<DeadEnd Up Right Down Left>; sub deq { defined$_[0] && $_[0] eq$_[1] }

sub perctest {
my($grid) = @_; generate($grid);
my $block = 1; for my$y (0..$grid-1) { for my$x (0..$grid-1) { fill($x, $y,$block++) if $perc[$y][$x] eq$fill
}
}
($block - 1) /$grid**2;
}

sub generate {
my($grid) = @_; for my$y (0..$grid-1) { for my$x (0..$grid-1) {$perc[$y][$x] = rand() < .5 ? '.' : $fill; } } } sub fill { my($x, $y,$block) = @_;
$perc[$y][$x] =$block;
my @stack;
while (1) {
if (my $dir = direction($x, $y )) { push @stack, [$x, $y]; ($x,$y) = move($dir, $x,$y, $block) } else { return unless @stack; ($x,$y) = @{pop @stack}; } } } sub direction { my($x, $y) = @_; return$D{Down}  if deq($perc[$y+1][$x ],$fill);
return $D{Left} if deq($perc[$y ][$x-1], $fill); return$D{Right} if deq($perc[$y  ][$x+1],$fill);
return $D{Up} if deq($perc[$y-1][$x  ], $fill); return$D{DeadEnd};
}

sub move {
my($dir,$x,$y,$block) = @_;
$perc[--$y][   $x] =$block if $dir ==$D{Up};
$perc[++$y][   $x] =$block if $dir ==$D{Down};
$perc[$y][ --$x] =$block if $dir ==$D{Left};
$perc[$y][ ++$x] =$block if $dir ==$D{Right};
($x,$y)
}

my $K = perctest(15); for my$row (@perc) {
printf "%3s", $_ for @$row;
print "\n";
}
printf  "𝘱 = 0.5, 𝘕 = 15, 𝘒 = %.4f\n\n", $K;$trials = 5;
for $N (10, 30, 100, 300, 1000) { my$total = 0;
$total += perctest($N) for 1..$trials; printf "𝘱 = 0.5, trials =$trials, 𝘕 = %4d, 𝘒 = %.4f\n", $N,$total / $trials; }  Output:  1 1 1 . . . . 2 2 2 . . . . . . 1 . 1 1 1 . 2 2 . 2 2 2 . 3 . 1 . . 1 . 2 2 2 2 2 2 . . 3 1 1 1 . 1 . 2 2 . . . . 4 4 . 1 1 1 . 1 . . 2 . . . . . . 1 1 1 1 1 1 . . 2 . . 5 . 6 . . 1 1 . . 1 1 . 2 . 7 . . . 1 1 1 . . . 1 1 . 2 2 . . 8 8 . 1 . 9 9 9 . 1 . . 2 2 . . . 1 1 . . 9 9 . . 10 . . . 11 . 12 . . . 9 9 . 13 13 . 13 . 14 . . 12 . . 15 . . 13 13 13 13 13 . . . 16 . 17 . 15 . . 13 . 13 . 13 13 . . 16 16 . . . 18 . . 13 13 13 13 . . . . . 19 19 1 . 1 . . 13 . . . . 20 . 19 19 . 𝘱 = 0.5, 𝘕 = 15, 𝘒 = 0.0889 𝘱 = 0.5, trials = 5, 𝘕 = 10, 𝘒 = 0.0980 𝘱 = 0.5, trials = 5, 𝘕 = 30, 𝘒 = 0.0738 𝘱 = 0.5, trials = 5, 𝘕 = 100, 𝘒 = 0.0670 𝘱 = 0.5, trials = 5, 𝘕 = 300, 𝘒 = 0.0660 𝘱 = 0.5, trials = 5, 𝘕 = 1000, 𝘒 = 0.0661 ## Phix Translation of: C with javascript_semantics sequence grid integer w, ww procedure make_grid(atom p) ww = w*w grid = repeat(0,ww) for i=1 to ww do grid[i] = -(rnd()<p) end for end procedure constant alpha = "+.ABCDEFGHIJKLMNOPQRSTUVWXYZ"& "abcdefghijklmnopqrstuvwxyz" procedure show_cluster() for i=1 to ww do integer gi = grid[i]+2 grid[i] = iff(gi<=length(alpha)?alpha[gi]:'?') end for puts(1,join_by(grid,w,w,"")) end procedure procedure recur(integer x, v) if x>=1 and x<=ww and grid[x]==-1 then grid[x] = v recur(x-w, v) recur(x-1, v) recur(x+1, v) recur(x+w, v) end if end procedure function count_clusters() integer cls = 0 for i=1 to ww do if grid[i]=-1 then cls += 1 recur(i, cls) end if end for return cls end function function tests(int n, atom p) atom k = 0 for i=1 to n do make_grid(p) k += count_clusters()/ww end for return k / n end function procedure main() w = 15 make_grid(0.5) printf(1,"width=15, p=0.5, %d clusters:\n", count_clusters()) show_cluster() printf(1,"\np=0.5, iter=5:\n") w = 4 while w<=4096 do printf(1,"%5d %9.6f\n", {w, tests(5,0.5)}) w *= 4 end while end procedure main()  Output: width=15, p=0.5, 18 clusters: ..EE.FF...OO..P .......KK.OOO.P A..B..I....OO.P .BBB.G.LL....PP BBB...J....J.P. ..BB..JJJJJJ.P. .BBBB.JJJ.J.J.. BBB....JJJ.JJ.Q BB.D.D.J.JJJ.QQ ..DDDD.JJ.JJJ.Q .DDDD.JJ..JJ... C.DDD.J.N.JJ... C.D...J..JJ.... .....H...J..... .......M..O...R p=0.5, iter=5: 4 0.137500 16 0.080469 64 0.068164 256 0.066809 1024 0.066018 4096 0.065777  ## Python from __future__ import division from random import random import string from math import fsum n_range, p, t = (2**n2 for n2 in range(4, 14, 2)), 0.5, 5 N = M = 15 NOT_CLUSTERED = 1 # filled but not clustered cell cell2char = ' #' + string.ascii_letters def newgrid(n, p): return [[int(random() < p) for x in range(n)] for y in range(n)] def pgrid(cell): for n in range(N): print( '%i) ' % (n % 10) + ' '.join(cell2char[cell[n][m]] for m in range(M))) def cluster_density(n, p): cc = clustercount(newgrid(n, p)) return cc / n / n def clustercount(cell): walk_index = 1 for n in range(N): for m in range(M): if cell[n][m] == NOT_CLUSTERED: walk_index += 1 walk_maze(m, n, cell, walk_index) return walk_index - 1 def walk_maze(m, n, cell, indx): # fill cell cell[n][m] = indx # down if n < N - 1 and cell[n+1][m] == NOT_CLUSTERED: walk_maze(m, n+1, cell, indx) # right if m < M - 1 and cell[n][m + 1] == NOT_CLUSTERED: walk_maze(m+1, n, cell, indx) # left if m and cell[n][m - 1] == NOT_CLUSTERED: walk_maze(m-1, n, cell, indx) # up if n and cell[n-1][m] == NOT_CLUSTERED: walk_maze(m, n-1, cell, indx) if __name__ == '__main__': cell = newgrid(n=N, p=0.5) print('Found %i clusters in this %i by %i grid\n' % (clustercount(cell), N, N)) pgrid(cell) print('') for n in n_range: N = M = n sim = fsum(cluster_density(n, p) for i in range(t)) / t print('t=%3i p=%4.2f n=%5i sim=%7.5f' % (t, p, n, sim))  Output: Found 20 clusters in this 15 by 15 grid 0) a a b c d d d d 1) a a e f g g d 2) e f f f d 3) h h e f i i d d 4) e j d d d d 5) k k k k l d d 6) k k k k k k l m n 7) k k k k k l o p p 8) k k k k l l l q 9) k k k k k l q q q 0) k k k k l q q q 1) k k k k k r r 2) k k k r r r s s 3) k k k k r r r r r s s 4) k k t r r r s s t= 5 p=0.50 n= 16 sim=0.08984 t= 5 p=0.50 n= 64 sim=0.07310 t= 5 p=0.50 n= 256 sim=0.06706 t= 5 p=0.50 n= 1024 sim=0.06612 t= 5 p=0.50 n= 4096 sim=0.06587 As n increases, the sim result gets closer to 0.065770... ## Racket #lang racket (require srfi/14) ; character sets ; much faster than safe fixnum functions (require racket/require ; for fancy require clause below (filtered-in (lambda (name) (regexp-replace #rx"unsafe-" name "")) racket/unsafe/ops) ; these aren't in racket/unsafe/ops (only-in racket/fixnum for/fxvector in-fxvector fxvector-copy)) ; ...(but less safe). if in doubt use this rather than the one above ; (require racket/fixnum) (define t (make-parameter 5)) (define (build-random-grid p M N) (define p-num (numerator p)) (define p-den (denominator p)) (for/fxvector #:length (fx* M N) ((_ (in-range (* M N)))) (if (< (random p-den) p-num) 1 0))) (define letters (sort (char-set->list (char-set-intersection char-set:letter ; char-set:ascii )) char<?)) (define n-letters (length letters)) (define cell->char (match-lambda (0 #\space) (1 #\.) (c (list-ref letters (modulo (- c 2) n-letters))))) (define (draw-percol-grid M N . gs) (for ((r N)) (for ((g gs)) (define row-str (list->string (for/list ((idx (in-range (* r M) (* (+ r 1) M)))) (cell->char (fxvector-ref g idx))))) (printf "|~a| " row-str)) (newline))) (define (count-clusters! M N g) (define (gather-cluster! k c) (when (fx= 1 (fxvector-ref g k)) (define k-r (fxquotient k M)) (define k-c (fxremainder k M)) (fxvector-set! g k c) (define-syntax-rule (gather-surrounds range? k+) (let ((idx k+)) (when (and range? (fx= 1 (fxvector-ref g idx))) (gather-cluster! idx c)))) (gather-surrounds (fx> k-r 0) (fx- k M)) (gather-surrounds (fx> k-c 0) (fx- k 1)) (gather-surrounds (fx< k-c (fx- M 1)) (fx+ k 1)) (gather-surrounds (fx< k-r (fx- N 1)) (fx+ k M)))) (define-values (rv _c) (for/fold ((rv 0) (c 2)) ((pos (in-range (fx* M N))) #:when (fx= 1 (fxvector-ref g pos))) (gather-cluster! pos c) (values (fx+ rv 1) (fx+ c 1)))) rv) (define (display-sample-clustering p) (printf "Percolation cluster sample: p=~a~%" p) (define g (build-random-grid p 15 15)) (define g+ (fxvector-copy g)) (define g-count (count-clusters! 15 15 g+)) (draw-percol-grid 15 15 g g+) (printf "~a clusters~%" g-count)) (define (experiment p n t) (printf "Experiment: ~a ~a ~a\t" p n t) (flush-output) (define sum-Cn (for/sum ((run (in-range t))) (printf "[~a" run) (flush-output) (define g (build-random-grid p n n)) (printf "*") (flush-output) (define Cn (count-clusters! n n g)) (printf "]") (flush-output) Cn)) (printf "\tmean K(p) = ~a~%" (real->decimal-string (/ sum-Cn t (sqr n)) 6))) (module+ main (t 10) (for ((n (in-list '(4000 1000 750 500 400 300 200 100 15)))) (experiment 1/2 n (t))) (display-sample-clustering 1/2)) (module+ test (define grd (build-random-grid 1/2 1000 1000)) (/ (for/sum ((g (in-fxvector grd)) #:when (zero? g)) 1) (fxvector-length grd)) (display-sample-clustering 1/2))  Output: Run from DrRacket, which runs the test and main modules. From the command line, you'll want two commands: racket percolation_m_c_d.rkt and raco test percolation_m_c_d.rkt for the same result. Experiment: 1/2 4000 10 [0*][1*][2*][3*][4*][5*][6*][7*][8*][9*] mean K(p) = 0.065860 Experiment: 1/2 1000 10 [0*][1*][2*][3*][4*][5*][6*][7*][8*][9*] mean K(p) = 0.066130 Experiment: 1/2 750 10 [0*][1*][2*][3*][4*][5*][6*][7*][8*][9*] mean K(p) = 0.066195 Experiment: 1/2 500 10 [0*][1*][2*][3*][4*][5*][6*][7*][8*][9*] mean K(p) = 0.066522 Experiment: 1/2 400 10 [0*][1*][2*][3*][4*][5*][6*][7*][8*][9*] mean K(p) = 0.066778 Experiment: 1/2 300 10 [0*][1*][2*][3*][4*][5*][6*][7*][8*][9*] mean K(p) = 0.066813 Experiment: 1/2 200 10 [0*][1*][2*][3*][4*][5*][6*][7*][8*][9*] mean K(p) = 0.067908 Experiment: 1/2 100 10 [0*][1*][2*][3*][4*][5*][6*][7*][8*][9*] mean K(p) = 0.069980 Experiment: 1/2 15 10 [0*][1*][2*][3*][4*][5*][6*][7*][8*][9*] mean K(p) = 0.089778 Percolation cluster sample: p=1/2 |. ... . . | |A BBB A A | |... .. .... | |AAA AA AAAA | |. . .... ... | |A A AAAA AAA | |. . . .........| |A A C AAAAAAAAA| | ... .. ....| | AAA AA AAAA| |.. ......... ..| |AA AAAAAAAAA AA| | . ... | | A AAA | |. .. .. | |D AA AA | | .. ... . .. | | AA AAA E AA | |. .. .. . . | |F AA AA A A | |. ........ . ..| |F AAAAAAAA A AA| |.. . .... ... | |FF A AAAA AAA | | . . . .... | | F G A AAAA | |.... .. .. . .| |FFFF HH AA A A| | . .. .....| | F HH AAAAA| 8 clusters ## Raku (formerly Perl 6) Works with: Rakudo version 2017.02 my @perc; my$fill = 'x';

enum Direction <DeadEnd Up Right Down Left>;

my $𝘒 = perctest(15); .fmt("%-2s").say for @perc; say "𝘱 = 0.5, 𝘕 = 15, 𝘒 =$𝘒\n";

my $trials = 5; for 10, 30, 100, 300, 1000 ->$𝘕 {
my $𝘒 = ( [+] perctest($𝘕) xx $trials ) /$trials;
say "𝘱 = 0.5, trials = $trials, 𝘕 =$𝘕, 𝘒 = $𝘒"; } sub infix:<deq> ($a, $b ) {$a.defined && ($a eq$b) }

sub perctest ( $grid ) { generate$grid;
my $block = 1; for ^$grid X ^$grid -> ($y, $x) { fill( [$x, $y],$block++ ) if @perc[$y;$x] eq $fill } ($block - 1) / $grid²; } sub generate ($grid ) {
@perc = ();
@perc.push: [ ( rand < .5 ?? '.' !! $fill ) xx$grid ] for ^$grid; } sub fill ( @cur,$block ) {
@perc[@cur[1]; @cur[0]] = $block; my @stack; my$current = @cur;

loop {
if my $dir = direction($current ) {
@stack.push: $current;$current = move $dir,$current, $block } else { return unless @stack;$current = @stack.pop
}
}

sub direction( [$x,$y] ) {
( Down  if @perc[$y + 1][$x] deq $fill ) || ( Left if @perc[$y][$x - 1] deq$fill ) ||
( Right if @perc[$y][$x + 1] deq $fill ) || ( Up if @perc[$y - 1][$x] deq$fill ) ||
}

sub move ( $dir, @cur,$block ) {
my ( $x,$y ) = @cur;
given $dir { when Up { @perc[--$y; $x] =$block }
when Down  { @perc[++$y;$x] = $block } when Left { @perc[$y; --$x] =$block }
when Right { @perc[$y; ++$x] = $block } } [$x, $y] } }  Output: . . 1 . 2 . . 3 . . . 4 . . . 2 2 . . 2 2 2 . 5 5 . 4 4 4 4 2 2 2 2 2 . 2 . 5 . . 4 . . 4 2 . 2 . 2 2 . . . . 4 4 4 . 4 . . . . . 2 . . . . 4 4 4 . . 6 6 6 6 . . 7 7 . . 4 . 4 4 . 6 . 6 . . . . . 4 4 4 4 4 . . 6 6 6 . . . 8 8 . 4 4 4 . . 4 6 . . . 9 . . . . . . 4 4 . 4 . 10 . 11 . . 12 12 . 4 . . 4 4 4 11 . 11 11 11 11 . 12 . 4 . 4 4 4 . 11 11 11 11 11 11 11 . . 4 4 . . 4 . 11 11 11 11 . 11 . . . 4 4 4 4 4 . 11 11 11 . 11 11 11 . . 4 4 4 4 . 13 . 11 11 . 11 11 . . . . 4 4 . 14 . 𝘱 = 0.5, 𝘕 = 15, 𝘒 = 0.062222 𝘱 = 0.5, trials = 5, 𝘕 = 10, 𝘒 = 0.114 𝘱 = 0.5, trials = 5, 𝘕 = 30, 𝘒 = 0.082444 𝘱 = 0.5, trials = 5, 𝘕 = 100, 𝘒 = 0.06862 𝘱 = 0.5, trials = 5, 𝘕 = 300, 𝘒 = 0.066889 𝘱 = 0.5, trials = 5, 𝘕 = 1000, 𝘒 = 0.0659358  ## Tcl Note that the queue (variables q and k) used to remember where to find cells when flood-filling the cluster is maintained as a list segment; the front of the list is not trimmed for performance reasons. (This would matter with very long queues, in which case the queue could be shortened occasionally; frequent trimming is still slower though, because Tcl backs its “list” datatype with arrays and not linked lists.) Works with: Tcl version 8.6 package require Tcl 8.6 proc determineClusters {w h p} { # Construct the grid set grid [lrepeat$h [lrepeat $w 0]] for {set i 0} {$i < $h} {incr i} { for {set j 0} {$j < $w} {incr j} { lset grid$i $j [expr {rand() <$p ? -1 : 0}]
}
}
# Find (and count) the clusters
set cl 0
for {set i 0} {$i <$h} {incr i} {
for {set j 0} {$j <$w} {incr j} {
if {[lindex $grid$i $j] == -1} { incr cl for {set q [list$i $j];set k 0} {$k<[llength $q]} {incr k} { set y [lindex$q $k] set x [lindex$q [incr k]]
if {[lindex $grid$y $x] != -1} continue lset grid$y $x$cl
foreach dx {1 0 -1 0} dy {0 1 0 -1} {
set nx [expr {$x+$dx}]
set ny [expr {$y+$dy}]
if {
$nx >= 0 &&$ny >= 0 && $nx <$w && $ny <$h &&
[lindex $grid$ny $nx] == -1 } then { lappend q$ny $nx } } } } } } return [list$cl $grid] } # Print a sample 15x15 grid lassign [determineClusters 15 15 0.5] n g puts "15x15 grid, p=0.5, with$n clusters"
puts "+[string repeat - 15]+"
foreach r $g {puts |[join [lmap x$r {format %c [expr {$x==0?32:64+$x}]}] ""]|}
puts "+[string repeat - 15]+"

# Determine the densities as the grid size increases
puts "p=0.5, iter=5"
foreach n {5 30 180 1080 6480} {
set tot 0
for {set i 0} {$i < 5} {incr i} { lassign [determineClusters$n $n 0.5] nC incr tot$nC
}
puts "n=$n, K(p)=[expr {$tot/5.0/$n**2}]" }  Output: 15x15 grid, p=0.5, with 21 clusters +---------------+ | A B CCCCC| | D A BBB C | |E B F CCCC| | B B F CC C| |BBB B BB CCC| |B BBBBBB CCCCC| | B B G C C| |H II G G J | |HH II G GG K| |HH II GGG GG K| | I G GGGG | |LL GGG GG M N| | L G G O P | |LLLL Q R | |L L S T UUU| +---------------+ p=0.5, iter=5 n=5, K(p)=0.184 n=30, K(p)=0.07155555555555557 n=180, K(p)=0.06880246913580246 n=1080, K(p)=0.0661267146776406 n=6480, K(p)=0.06582889898643499  ## Wren Translation of: Kotlin Library: Wren-fmt import "random" for Random import "./fmt" for Fmt var rand = Random.new() var RAND_MAX = 32767 var list = [] var w = 0 var ww = 0 var ALPHA = "+.ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz" var ALEN = ALPHA.count - 3 var makeList = Fn.new { |p| var thresh = (p * RAND_MAX).truncate ww = w * w var i = ww list = List.filled(i, 0) while (i != 0) { i = i - 1 var r = rand.int(RAND_MAX+1) if (r < thresh) list[i] = -1 } } var showCluster = Fn.new { var k = 0 for (i in 0...w) { for (j in 0...w) { var s = list[k] k = k + 1 var c = (s < ALEN) ? ALPHA[1 + s] : "?" System.write(" %(c)") } System.print() } } var recur // recursive recur = Fn.new { |x, v| if (x >= 0 && x < ww && list[x] == -1) { list[x] = v recur.call(x - w, v) recur.call(x - 1, v) recur.call(x + 1, v) recur.call(x + w, v) } } var countClusters = Fn.new { var cls = 0 for (i in 0...ww) { if (list[i] == -1) { cls = cls + 1 recur.call(i, cls) } } return cls } var tests = Fn.new { |n, p| var k = 0 for (i in 0...n) { makeList.call(p) k = k + countClusters.call() / ww } return k / n } w = 15 makeList.call(0.5) var cls = countClusters.call() System.print("width = 15, p = 0.5, %(cls) clusters:") showCluster.call() System.print("\np = 0.5, iter = 5:") w = 1 << 2 while (w <= (1 << 13)) { var t = tests.call(5, 0.5) Fmt.print("$5d \$9.6f", w, t)
w = w << 1
}

Output:

Sample run:

width = 15, p = 0.5, 28 clusters:
. A A . . B . C . D . . . . A
A A A . E . F . G . . A A A A
. A A A . F F . . . A A . . .
. A . . . . F . . H . A . A .
. A A . . . . . H H H . A A A
A A A . . I I . . H . J . . A
A . . K . . . . H H . . . A A
A . L . M M . N . . O . . . .
A A . P . . . N . Q . R . S .
. A A . T T . . U . R R R . .
A A . . T T T . . . . R . . .
. A . T T T . . . . . R R R R
. A . . . T . . V . W . R R .
A A . X . T T . . . W . . . .
. . Y . T T . Z . a . . . b b

p = 0.5, iter = 5:
4  0.125000
8  0.081250
16  0.094531
32  0.073242
64  0.067920
128  0.068567
256  0.067404
512  0.066401
1024  0.065870
2048  0.065885
4096  0.065812
8192  0.065795


## zkl

Translation of: C
const X=-1;	// the sentinal that marks an untouched cell
var C,N,NN,P;
fcn createC(n,p){
N,P=n,p; NN=N*N;
C=NN.pump(List.createLong(NN),0);  // vector of ints
foreach n in (NN){ C[n]=X*(Float.random(1)<=P) } // X is the sentinal
}
fcn showCluster{
alpha:="-ABCDEFGHIJKLMNOPQRSTUVWXYZ" "abcdefghijklmnopqrstuvwxyz";
foreach n in ([0..NN,N]){ C[n,N].pump(String,alpha.get).println() }
}
fcn countClusters{
clusters:=0;
foreach n in (NN){
if(X!=C[n]) continue;
fcn(n,v){
if((0<=n<NN) and C[n]==X){
C[n]=v;
self.fcn(n-N,v); self.fcn(n-1,v); self.fcn(n+1,v); self.fcn(n+N,v);
}
}(n,clusters+=1);
}
clusters
}
fcn tests(N,n,p){
k:=0.0;
foreach z in (n){ createC(N,p); k+=countClusters().toFloat()/NN; }
k/n
}
createC(15,0.5);
println("width=%d, p=%.1f, %d clusters:".fmt(N,P,countClusters()));
showCluster();

println("p=0.5, 5 iterations:");
w:=4; do(6){ println("%5d %9.6f".fmt(w,tests(w, 5, 0.5))); w*=4; }
Output:
width=15, p=0.5, 16 clusters:
-AAA-BB-BBB---C
------BBBB--D--
E---F---BB--DD-
EE----G-BB---DD
--H-I--J--J--DD
-K--I--JJ-J--D-
-K--I--JJJJ-L--
KK-III-------MM
-K-I--I--NN-I--
I-IIIII-NNN-III
I-II--I-N-N-II-
III-III--NNN-II
I-II-II-O---I--
I-I-IIII-PP-III
I-II--I---P--II

p=0.5, 5 iterations:
4  0.062500
16  0.070312
64  0.067627
256  0.067078
1024  0.065834
4096  0.065771