Agglomerative percolation on the Bethe lattice and the triangular cactus

TL;DR

This paper develops an exact mean-field theory for agglomerative percolation on the Bethe lattice and triangular cactus, measuring critical exponents and confirming a new universality class depending on symmetry breaking.

Contribution

It provides the first exact mean-field analysis of agglomerative percolation on these structures and characterizes the critical exponents for different symmetry breakings.

Findings

Critical exponents for $k=2$ and 3 are measured.

The exponents differ from ordinary percolation, indicating a new universality class.

The exponents follow specific inequalities, confirming theoretical predictions.

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 $\beta_k$ and $\gamma_k$ for $k=2$ and 3. Here $\beta_k$ and $\gamma_k$ 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 $\beta_2=1.79(3)$, $\gamma_2=0.88(1)$, $\beta_3=1.35(5)$, and $\gamma_3=0.94(2)$. By comparing these values of exponents with those for ordinary percolation ($\beta_{\infty}=1$ and $\gamma_{\infty}=1$) we also find the inequalities between the exponents, as $\beta_\infty<\beta_3<\beta_2$ and $\gamma_\infty>\gamma_3>\gamma_2$. 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)].