Renormalization-group theory of the abnormal singularities at the critical-order transition in bond percolation on pointed hierarchical graphs

TL;DR

This paper develops a renormalization-group framework to analyze the singularities at phase transitions in bond percolation on pointed hierarchical graphs, revealing how bifurcation types influence the order parameter's critical behavior.

Contribution

It establishes a direct relation between RG bifurcation types and the singularity of the order parameter in hierarchical graph percolation models, including conditions for power-law and discontinuous transitions.

Findings

Bifurcation type determines the singularity of the order parameter.

Power-law or essential singularities depend on the local graph structure.

First-order perturbation effects vanish under specific graph connectivity conditions.

Abstract

We study the singularity of the order parameter at the transition between a critical phase and an ordered phase of bond percolation on pointed hierarchical graphs. In pointed hierarchical graphs, the renormalization group (RG) equation explicitly depends on the bare parameter, which causes the phase transitions that correspond to the bifurcation of the RG fixed point. We derive the relation between the type of this bifurcation and the type of the singularity of the order parameter. In the case of a saddle node bifurcation, the singularity of the order parameter is power-law or essential one depending on the fundamental local structure of the graph. In the case of pitchfork and transcritical bifurcations, the singularity is essential and power-law ones, respectively. These becomes power-law and discontinuous ones, respectively, in the absence of the first-order perturbation to the largest eigenvalue of the combining matrix, which gives the growth rate of the cluster size. We also show that the first-order perturbation vanishes if the relevant RG parameter is unique and the backbone of the pointed hierarchical graph is simply connected via nesting subunits.