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Transition to Percolation

Understanding the Percolation Transition

In one-dimensional percolation, the transition to a percolating state is particularly sharp. The probability that a site belongs to the percolating infinite cluster, denoted as P∞(p), exhibits a discontinuous jump at p = 1:

$$P_\infty(p) = \begin{cases} 0 & \text{for } p < 1 \\ 1 & \text{for } p = 1 \end{cases}$$

This means that a percolating cluster only exists when every single site is occupied - even a single gap would break the connection across the system.

Conservation of Probability

The total probability of site occupation can be written as a sum of two terms:

$$P_\infty(p) + \sum_{s=1}^\infty sn(s,p) = p$$

This equation expresses that an occupied site must either belong to:

  • The infinite percolating cluster (P∞(p))
  • A finite cluster (Σsn(s,p))

Interactive 1D Percolation

p = 0.800

Status: Not Percolating

In 1D, percolation only occurs when all sites are occupied (p = 1). Even a single gap prevents percolation.

Probability Distribution

The plot shows:

  • P∞(p): Probability of belonging to infinite cluster (orange)
  • Σsn(s,p): Probability of belonging to finite clusters (purple)

Notice the sharp transition at p = 1, where P∞(p) jumps from 0 to 1, and correspondingly, the sum over finite clusters drops to 0.