Agglomerative percolation on the Bethe lattice and the triangular cactus

Agglomerative percolation (AP) on the Bethe lattice and the triangular cactus is studied to establish the exact mean-field theory for AP. Using the self-consistent simulation method based on the exact self-consistent equations, the order parameter P_∞ and the average cluster size S are measured. From the measured P_∞ and S, the critical exponents β_k and γ_k for k = 2 and 3 are evaluated. Here, β_k and γ_k are the critical exponents for P _∞ and S when the growth of clusters spontaneously breaks the Z_k symmetry of the k-partite graph. The obtained values are β₂ = 1.79(3), γ₂ = 0.88(1), β₃ = 1.35(5) and γ₃ = 0.94(2). By comparing these exponents with those for ordinary percolation (β_∞ = 1 and γ_∞ = 1), we also find β_∞ < β₃ < β₂ and γ_∞ > γ₃ > γ₂. These results quantitatively verify the conjecture that the AP model belongs to a new universality class if the Z_k symmetry is broken spontaneously, and the new universality class depends on k.