Step by Step solution about How to resolve the algorithm Percolation/Mean run density step by step in the Java programming language

The given Java code is intended to explore the behavior of a percolation model and compare the empirical results with the theoretical expectations. A percolation model simulates the spread of a fluid through a porous medium by randomly opening or closing connections between points in a grid.

Here's a detailed explanation of the code:

1. Main Method: The program starts by running a series of tests with varying parameters and displaying the results in a tabular format.

2. Run Length: For each test, it varies the run length (length) from 100 to 100,000 in powers of 10.

3. Probability of Opening: The probability of opening a connection between two points (probability) is varied from 0.1 to 0.9 in increments of 0.2.

4. Theoretical Expectation: For each combination of probability and length, the theoretical probability of having a continuous path from top to bottom in the grid is calculated using the formula theory = probability * ( 1.0 - probability ).

5. Empirical Measurement: The runTest method is called to perform multiple runs of the percolation experiment for a given probability and length. Each run simulates a series of connected or disconnected points in a grid. The result is the fraction of runs that have a continuous path from top to bottom.

In each run:

• previousBit and nextBit represent the state of two consecutive points in the grid, initially set to 0.
• The grid is traversed by generating a random number for each point. If the random number is less than probability, the point is opened (nextBit set to 1), otherwise, it remains closed (nextBit set to 0).
• If the current point is opened and the previous point was closed, the counter count is incremented, indicating that a new continuous path has started.
• previousBit is updated to the value of nextBit for the next iteration.

6. Results Display: The empirical result (result) is calculated as count / aRunCount / aLength. The results are printed in a table with columns for probability, length, result, theory, and the difference between result and theory.

7. Thread Local Random: The random variable uses ThreadLocalRandom for generating random numbers, which provides faster and more efficient random number generation than the traditional Random class.

Overall, the code performs multiple simulations of a percolation model and compares the empirical measurements with theoretical expectations, showing how the probability of percolation changes with different parameters. The results are presented in a tabular format for easy comparison and analysis.

import java.util.concurrent.ThreadLocalRandom;

public final class PercolationMeanRun {

	public static void main(String[] aArgs) {
		System.out.println("Running 1000 tests each:" + System.lineSeparator());
		System.out.println(" p\tlength\tresult\ttheory\t   difference");
		System.out.println("-".repeat(48));
		
	    for ( double probability = 0.1; probability <= 0.9; probability += 0.2 ) {
	        double theory = probability * ( 1.0 - probability );
	        int length = 100;
	        while ( length <= 100_000 ) {
	            double result = runTest(probability, length, 1_000);
	            System.out.println(String.format("%.1f\t%6d\t%.4f\t%.4f\t%+.4f (%+.2f%%)",
	            	probability, length, result, theory, result - theory, ( result - theory ) / theory * 100));
	            length *= 10;
	        }
	        System.out.println();
	    }

	}
	
	private static double runTest(double aProbability, int aLength, int aRunCount) {
		double count = 0.0;
	    for ( int run = 0; run < aRunCount; run++ ) {
	        int previousBit = 0;
	        int length = aLength;
	        while ( length-- > 0 ) {
	            int nextBit = ( random.nextDouble(1.0) < aProbability ) ? 1 : 0;
	            if ( previousBit < nextBit ) {
	            	count += 1.0;
	            }
	            previousBit = nextBit;
	        }
	    }
	    return count / aRunCount / aLength;
	}

	private static ThreadLocalRandom random = ThreadLocalRandom.current();

}