Percolation: Clusters & Critical Threshold in 2D

What Percolation Is

Percolation is a process that occurs when fluid, particles, or other substances pass through a porous medium. In a 2D context, it refers to the formation of connected pathways for a fluid to flow through a network of randomly distributed obstacles.

The percolation model can be visualized as a grid where each site (or link) is either 'open' with probability p or 'closed' with probability 1-p. When enough sites are open, a continuous path from the top to the bottom of the grid forms.

Why It Happens

The phenomenon arises due to the random nature of the connections and how they cluster together as more links become 'open'. This clustering behavior is governed by the percolation threshold, which marks the point where a giant connected component first appears.

Above this critical probability, there exists a spanning cluster that spans from one side of the grid to the other, allowing for fluid or particles to pass through the entire system.

The Critical Threshold

The critical threshold (pc) is the specific value of p at which percolation occurs. Below this threshold, no spanning cluster exists; above it, a giant connected component emerges.

Understanding pc and how clusters form helps in various fields such as material science, network theory, and even social dynamics.

Real-World Applications

Percolation theory is crucial in studying the flow of fluids through porous materials like rocks or filters. It also applies to understanding electrical conductivity in disordered systems.

In network theory, percolation models help analyze the robustness and connectivity of networks such as power grids or the internet.

Frequently asked questions

Try it live

Everything above runs in your browser — open Percolation — Clusters & Critical Threshold (2D) and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.