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Introduction to Percolation Theory

Interactive Percolation Grid

Occupation Probability: 0.50No Percolation

Number of Clusters: 0

A cluster is a group of connected occupied sites (orange squares). Percolation occurs when there's a path from top to bottom.

Understanding Percolation

Percolation theory studies how connected regions form in random systems. Imagine taking a piece of paper and randomly coloring squares - at what point will a continuous path form from top to bottom?

Key Concepts:

  • Phase Transition: A critical probability where infinite clusters first appear
  • Clusters: Groups of connected occupied sites
  • Scale Invariance: Similar patterns at different scales near the critical point

Real-World Applications:

  • Oil recovery from porous media
  • Epidemic modeling
  • Network resilience
  • Forest fire spread
  • Groundwater flow

Try adjusting the probability slider to see how cluster formation changes. Notice how the system behaves differently above and below the critical probability (approximately 0.59 for a square lattice).