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Transition to Percolation

Order Parameter P∞(p)

The probability P∞(p) that a site belongs to the percolating infinite cluster serves as an order parameter for the percolation transition. For z = 3:

$$P_\infty(p) = \begin{cases} 0 & \text{for } p \leq p_c \\ p\left[1-\left(\frac{1-p}{p}\right)^3\right] & \text{for } p > p_c \end{cases}$$

Near the critical point, P∞(p) can be expanded in a Taylor series:

$$P_\infty(p) = 6(p-p_c) - 24(p-p_c)^2 + \cdots \quad \text{for } p > p_c$$

Key Features

  • • Continuous but non-differentiable at pc
  • • Zero below pc (no infinite cluster)
  • • Linear growth just above pc
  • • Universal behavior near pc

Physical Meaning

  • • Measures fraction in infinite cluster
  • • Marks continuous phase transition
  • • Shows emergence of macroscopic order
  • • Critical exponent β = 1 is universal