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Agglomerative percolation on the Bethe lattice and the triangular cactus

We study the agglomerative percolation (AP) models on the Bethe lattice and the triangular cactus to establish the exact mean-field theory for AP. Using the self-consistent simulation method, based on the exact self-consistent equation, we directly measure the order parameter $P_{\infty}$ and average cluster size $S$. From the measured $P_{\infty}$ and $S$ we obtain the critical exponents $β_k$ and $γ_k$ for $k=2$ and 3. Here $β_k$ and $γ_k$ are the critical exponents for $P_\infty$ and $S$ when the growth of clusters spontaneously breaks the $Z_k$ symmetry of the $k$-partite graph (Lau, Paczuski, and Grassberger, 2012). The obtained values are $β_2=1.79(3)$, $γ_2=0.88(1)$, $β_3=1.35(5)$, and $γ_3=0.94(2)$. By comparing these values of exponents with those for ordinary percolation ($β_{\infty}=1$ and $γ_{\infty}=1$) we also find the inequalities between the exponents, as $β_\infty<β_3<β_2$ and $γ_\infty>γ_3>γ_2$. These results quantitatively verify the conjecture that the AP model belongs to a new universality class if $Z_k$ symmetry is broken spontaneously, and the new universality class depends on $k$ [Lau et al., Phys. Rev. E 86, 011118 (2012)].