Percolation, explained.
Percolation studies connected paths in randomly open structures, including whether local open sites form a crossing through a larger system.
Why it happens
An open fraction is only part of the story. Sites must connect in the permitted directions, and a closed barrier can block passage even when many other sites are open.
Local connections can enable a system-spanning path. A finite random grid illustrates connectivity without guaranteeing a path at a particular density.
Read the result
Increase the opening probability while keeping the same seeded board. Blue tiles are connected to the top; rose tiles are open but unreachable. Resample to compare arrangements with the same opening chance.
A worked example
A porous crossing
Some tiles in a ten-by-ten board are open, but none connect the top to the bottom.
Opening one tile that joins two previously separate regions can complete a crossing.
A small local change may create global connectivity; the important tile depends on the surrounding arrangement.
OPTIONAL DEEPER DETAILGo deeper: inside the model
Inside this model
A fixed seeded 10×10 grid assigns a threshold to each tile. Increasing openness activates a nested set. Flood fill uses only up, down, left and right neighbors from open top-row tiles.
Where this idea is useful
A practical use
Connected gaps determine passage through porous material; the fraction of gaps alone does not describe their arrangement.
A common misconception
“The same open percentage guarantees the same connectivity.”
Different arrangements can have different paths and barriers. This small board cannot establish an infinite-grid threshold.
What this explanation leaves out
- This small site-percolation board does not estimate an infinite-lattice threshold or model real fluid dynamics.
Why do diagonal contacts not count?
The model defines connectivity through shared edges. Changing that rule changes the paths and the relevant percolation problem.
Which local connection would turn separate parts of your system into a functioning route?
Further reading
Explore the original research or the teaching reference behind this experiment.