Agglomerative percolation on the Bethe lattice and the triangular cactus

Abstract

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)].

13 pages. 6 figures