Phase transitions in the Ising model on a hierarchical random graph based on the triangle

Abstract

Hierarchical graphs were invented to formalize heuristic Migdal–Kadanoff renormalization arguments. In such graphs, certain characteristic patterns (motifs) appear as construction elements. Real-world complex networks may also contain such patterns. Itzkovitz and Alon in 2005 Phys. Rev. E 71, selected five most typical motifs, which include the triangle. In 2011 Condens. Matter. Phys. 14, Kotorowicz and Kozitsky introduced and described hierarchical random graphs in which these five motifs appear at each hierarchy level. In the present work, we study the equilibrium states of the Ising spin model living on the graph of this kind based on the triangle. The main result is the description of annealed phase transitions in this model. In particular, we show that—depending on the parameters—the model may be in an unordered or ordered states at all temperatures, as well as to have a critical point. The key aspect of our theory is detecting the appearance of an ordered state by the non-ergodicity of a certain nonhomogeneous Markov chain.