Mean-Field Percolation — Bethe Lattice

Bethe lattice percolation — The Bethe lattice (Cayley tree) with coordination number z gives the exact mean-field percolation solution. Bond probability p; critical threshold p_c = 1/(z−1).

The giant-component probability P∞ satisfies the self-consistency equation: P∞ = 1 − (1 − p·P∞)^(z−1) (solved iteratively). Near p_c: P∞ ~ (p − p_c)^β with β = 1 (mean-field).

Mean cluster size χ = (1 + p(z−1))/(1 − p(z−1)) diverges at p_c. Correlation length ξ ~ |p − p_c|^−ν, ν = 1/2 (mean-field critical exponent).

The tree visualization shows a random realization: open bonds (gold) vs closed (dark). The right panel plots P∞ and χ (normalized) vs p.