2017/10/31 by Tomoaki Nogawa
Mathematics · Physics and Astronomy · #Bifurcation #Complex Network Analysis Techniques #Eigenvalues and eigenvectors #Graph theory and applications #Gravitational singularity #Perturbation (astronomy) #Phase transition #Renormalization group #Saddle-node bifurcation #Singularity #Singularity theory #Theoretical and Computational Physics #cond-mat.dis-nn #cond-mat.stat-mech
paper · pdf · doi:10.1088/1751-8121/aae79e
17 pages, 3 figures
openalex created_date 2017/10/20 · openalex publication_date 2018/10/11 · arxiv created 2018/12/20 · arxiv updated 2018/12/21 · openalex updated_date 2026/08/05
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 (PHGs). In PHGs with shortcuts, the renormalization group (RG) equation explicitly depends on the bare parameter, which causes the phase transition that corresponds to the bifurcation of the RG fixed point. We derive the general 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 is a power–law or essential type depending on the fundamental local structure of the graph. In the case of pitchfork and transcritical bifurcations, the singularity is essential and power–law types, respectively. These are replaced by power–law and discontinuous types, 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 backbone of the PHG is simply connected via nesting subunits and all the roots of the PHG are almost surely connected in the ordered phase.