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

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.