Percolation Thresholds: When Random Dots Suddenly Connect

Percolation is the moment a random local rule becomes a system-level event. A single dot is boring. A random grid of dots is still simple. But near a critical probability, one more open site can turn disconnected islands into a spanning path.

The Critical Percolation Threshold Lab makes that transition inspectable. It draws the clusters, highlights the largest component, detects whether a path spans the grid and runs a finite-size Monte Carlo sweep around the chosen probability.

The Threshold Idea

For infinite square-lattice site percolation, the critical probability is approximately pc = 0.592746. Below that level, open sites usually form isolated islands. Above it, a connected component is much more likely to cross the system.

Why Finite Grids Need Diagnostics

A browser tool cannot honestly claim an infinite-lattice proof from one finite grid. The useful move is to show the grid result and a local sweep together. If span probability changes sharply around the selected p, the user can see the transition window instead of trusting a single picture.

What The Metrics Mean

Largest cluster ratio tells how much of the grid belongs to the biggest connected component. Span direction tells whether a component crosses top-to-bottom or left-to-right. Threshold gap compares the selected probability with the reference pc. Span probability estimates how often nearby runs cross the grid.

How To Use The Lab

1. Open the Critical Percolation Threshold Lab.
2. Run the near-critical sample.
3. Move p below 0.56 and watch clusters fragment.
4. Move p above 0.64 and watch spanning become common.
5. Use the sweep CSV when writing or teaching the transition.

What This Does Not Prove

This is an educational finite-grid model. It is not infrastructure certification, safety engineering, medical spread modeling, legal advice, financial advice, traffic proof, ranking proof or revenue proof. Its value is that it makes the threshold visible and measurable without pretending to certify the real world.