Bethe Lattice Percolation — Mean-Field Critical Point

Bond probability p 0.40

Branching z 3

Depth levels 5

p_c = 1/(z−1) 0.500

Giant cluster P∞ —

Mean cluster ⟨s⟩ —

On the Bethe lattice (infinite Cayley tree with branching z), percolation is exactly solvable: p_c = 1/(z−1). Near the critical point, the infinite cluster probability P∞ ~ (p−p_c)^β with β=1 (mean-field). The mean cluster size ⟨s⟩ ~ |p−p_c|^{−γ} with γ=1. These are the mean-field exponents — exact for trees, and the upper critical dimension d≥6 for lattices.