Nogawa, Tomoaki Renormalization-group theory of the abnormal singularities at the critical-order transition in bond percolation on pointed hierarchical graphs. (English) Zbl 1411.82019 J. Phys. A, Math. Theor. 51, No. 50, Article ID 505003, 17 p. (2018). Summary: 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. MSC: 82B26 Phase transitions (general) in equilibrium statistical mechanics 82B27 Critical phenomena in equilibrium statistical mechanics 82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics 82B43 Percolation 82B28 Renormalization group methods in equilibrium statistical mechanics 37G10 Bifurcations of singular points in dynamical systems 05C82 Small world graphs, complex networks (graph-theoretic aspects) Keywords:critical phenomena; percolation; renormalization group; complex networks PDFBibTeX XMLCite \textit{T. Nogawa}, J. Phys. A, Math. Theor. 51, No. 50, Article ID 505003, 17 p. (2018; Zbl 1411.82019) Full Text: DOI arXiv References: [1] Dorogovtsev, S. N.; Goltsev, A. V.; Mendes, J. F F., Rev. Mod. Phys., 80, 1275, (2008) [2] Ravasz, E.; Barabási, A. L., Phys. Rev. E, 67, (2003) [3] Hasegawa, T.; Nogawa, T.; Nemoto, K., Discontinuity Nonlinearity Complexity, 3, 319, (2014) [4] Berezinskii, V. L., Zh. Eksp. Teor. Fiz., 61, 1144, (1972) [5] Kosterlitz, J. M.; Thouless, D. J., J. Phys. C: Solid State Phys., 6, 1181, (1973) [6] Kosterlitz, J. M., J. Phys. C: Solid State Phys., 7, 1046, (1974) [7] Boettcher, S.; Brunson, T., Europhys. Lett., 110, 26005, (2015) [8] Hinczewski, M.; Berker, A. N., Phys. Rev. E, 73, (2006) · Zbl 1244.82013 [9] Chui, S. T.; Weeks, J. D., Phys. Rev. B, 14, 4978, (1976) [10] Jose, J. V.; Kadanoff, L. P.; Kirkpatric, S.; Nelson, D. R., Phys. Rev. B, 16, 1217, (1977) [11] Krapivsky, P. L.; Derrida, B., Physica A, 340, 714, (2004) [12] Bollobás, B.; Riordan, O., Random Struct. Algorithms, 27, 1, (2005) · Zbl 1074.05081 [13] Riordan, O., Comb. Probab. Comput., 14, 897, (2005) · Zbl 1079.05092 [14] Berker, A. N.; Hinczewski, M.; Netz, R. R., Phys. Rev. E, 80, (2009) [15] Hasegawa, T.; Sato, M.; Nemoto, K., Phys. Rev. E, 82, (2010) [16] Boettcher, S.; Cook, J. L.; Ziff, R. M., Phys. Rev. E, 80, (2009) [17] Hasegawa, T.; Nemoto, K., Phys. Rev. E, 81, (2010) [18] Bauer, M.; Coulomb, S.; Dorogovtsev, S. N., Phys. Rev. Lett., 94, (2005) [19] Boettcher, S.; Brunson, C. T., Phys. Rev. E, 83, (2011) [20] Nogawa, T.; Hasegawa, T.; Nemoto, K., Phys. Rev. Lett., 108, (2012) [21] Nogawa, T.; Hasegawa, T.; Nemoto, K., Phys. Rev. E, 86, (2012) [22] Boettcher, S.; Singh, V.; Ziff, R. M., Nat. Commun., 3, 787, (2012) [23] Nogawa, T.; Hasegawa, T., J. Phys. A: Math. Theor., 42, (2009) · Zbl 1160.82327 [24] Eggarter, T. P., Phys. Rev. B, 9, 2989, (1974) [25] Müller-Hartmann, E.; Zittartz, J., Phys. Rev. Lett., 33, 893, (1974) [26] Ostilli, M., Physica A, 391, 3417, (2012) [27] Nogawa, T.; Hasegawa, T.; Nemoto, K., J. Stat. Mech., (2016) [28] Nogawa, T.; Hasegawa, T., Phys. Rev. E, 89, (2014) [29] Zhang, Z.; Comellas, F., Theor. Comput. Sci., 412, 865, (2011) · Zbl 1206.68245 [30] Rozenfeld, H. D.; ben Avraham, D., Phys. Rev. E, 75, (2007) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.