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Critical Behavior in Bethe Lattices

Critical Occupation Probability

In a Bethe lattice, the critical occupation probability depends on the coordination number z. For a walk on the percolating cluster to continue indefinitely, at least one branch must be accessible at each step:

$$p(z-1) \sim 1$$$$p_c = \frac{1}{z-1}$$
Coordination number (z): 3

Current pc = 0.500

Average Cluster Size

The average cluster size χ(p) in a Bethe lattice can be calculated exactly:

$$\chi(p) = \frac{p_c(1+p)}{p_c-p} \quad \text{for } 0 < p < p_c$$

Key Properties:

  • χ(p) diverges as p approaches pc from below
  • Power law divergence with exponent -1
  • Recovers 1D behavior when z = 2
  • Shows mean-field behavior for large z

Universal Power Law Behavior

As p approaches pc from below, the average cluster size follows a remarkable universal power law:

$$\chi(p) = \frac{p_c(1+p)}{p_c-p} \to p_c(1+p_c)(p_c-p)^{-1} \quad \text{for } p \to p_c^-$$

Understanding the Plot

  • • X-axis shows distance from pc (log scale)
  • • Y-axis shows average cluster size (log scale)
  • • Straight lines indicate power law behavior
  • • Parallel slopes confirm universal exponent

Physical Significance

  • • Universal behavior suggests fundamental principle
  • • Different pc values show lattice dependence
  • • Common exponent transcends lattice details
  • • Similar behavior in many physical systems

This universality is a profound feature of critical phenomena. While the location of the critical point (pc) depends on microscopic details like the coordination number, the nature of the divergence (exponent -1) is universal. This suggests that the behavior near criticality is governed by more fundamental principles that transcend the specific details of the system.