Critical Behavior and Bethe Lattices

Critical Occupation Probability

In one-dimensional percolation, the critical occupation probability pc = 1 marks a special point where several quantities diverge simultaneously:

$$s_\xi(p) \sim (1-p)^{-1}$$$$\chi(p) \sim (1-p)^{-1}$$$$\xi(p) \sim (1-p)^{-1}$$

Bethe Lattice (Cayley Tree)

The Bethe lattice, also known as a Cayley tree, is a special mathematical structure that provides an exactly solvable model for percolation. It has several unique properties that make it valuable for theoretical analysis:

Each site has exactly z neighbors (except at the boundary)
The structure contains no loops or cycles
All paths between two points are unique
The number of sites grows exponentially with distance from center

Coordination number (z): 3

Critical Properties:

Critical probability: $p_c = \frac{1}{z-1}$
Current pc for z = 3: 0.500
As z → ∞, pc → 0, approaching mean-field behavior

Growth Properties:

Number of sites at level n: $N_n = z(z-1)^{n-1}$
Total sites up to level n: $N_{total} = 1 + \sum_{k=1}^n z(z-1)^{k-1}$
Branching ratio: $b = z-1$

Significance in Percolation Theory:

Provides exact solutions due to its loop-free structure
Bridges between 1D behavior (z=2) and infinite-dimensional behavior (z→∞)
Demonstrates mean-field critical behavior for large z
Useful for understanding phase transitions in complex networks
Shows clear distinction between subcritical and supercritical phases

Note: While real physical systems rarely have the exact structure of a Bethe lattice, its analysis provides valuable insights into critical phenomena and phase transitions. Many real networks locally resemble a Bethe lattice, making it a useful approximation for understanding more complex systems.