Percolation
Percolation asks a simple question with a sharp answer: if you open each cell of a grid independently with probability p, when does an open path first reach all the way across? Drag p from 0 to 1 and the tool fills the grid, floods the connected clusters in colour and highlights any cluster that spans top to bottom. Below a critical p the clusters stay small and isolated; cross it and a giant cluster suddenly appears and the grid conducts. This threshold is startlingly abrupt for something built from independent coin flips, and the tool marks the spanning transition as you sweep p.
Runs 100% in your browser — simulations are computed locally on your device.
Read the full guide to this tool
Notes
- Each site is open independently with probability p, so there is no coordination anywhere — yet a system-spanning cluster appears suddenly at a critical p, near 0.5927 for site percolation on a square grid.
- Below the threshold the largest cluster is tiny compared with the grid; above it, one cluster fills a finite fraction of the whole — the same abrupt jump seen in phase transitions.
- Percolation models many real spreading problems: current through a random conductor, water through soil, fire through a forest and disease through a contact network.
- Runs 100% in your browser — simulations are computed locally on your device.
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