When does a maze suddenly become passable? Raise the fraction of open cells and watch a connected path snap into existence at the critical threshold.
Percolation StudioLive
phase transition · spanning clusters
Controls
Each cell is open with probability p. Water poured on top (cyan) seeps through connected open cells. Near p ≈ 0.59 a spanning path suddenly appears — a phase transition.
Reading this result: Right at the critical point p_c ≈ 0.5927: a spanning cluster first appears. This is a continuous (second-order) phase transition — the crossing probability rises sharply here and the connected cluster is fractal.
Runs locally in your browser — free forever. Scale to the cloud when reality gets heavy.
★ Sign in to save this setupSave your tuned setup, or drop this simulation into your own site, docs, or course page.
How it works
In site percolation each cell is open with probability p. Fluid poured on top flows through connected open cells. Below the critical probability (≈ 0.593 for a square lattice) it never reaches the bottom; above it, a spanning cluster appears almost surely — a sharp phase transition central to materials science and network theory.
✦
Ask the AI about this model
The math, the assumptions, real-world uses, or a code translation — explained for this exact simulation.
Yes. Percolation runs entirely in your browser using your device's own compute, so local use is free forever. You only pay Compute Tokens if you scale a job to the cloud.
Do I need to install anything?▾
No. Everything runs client-side in a modern browser — no downloads, no license, no account required to start.
Can I save or share my simulation?▾
Create a free account to save projects, and use a shareable embed or minted DOI to publish a live, interactive version anywhere.
How accurate are the results?▾
The solver uses established numerical methods, but results are for research and educational purposes and should be validated against experiment or professional review before you rely on them.