Stefan Boettcher, Trent Brunson
A generic classification of critical behavior is provided in systems for which the renormalization group equations are parameter dependent. It describes phase transitions in a wide class of networks with a hierarchical structure but also applies, for instance, to conformal field theories. Although these transitions generally do not exhibit universality, three distinct regimes of characteristic critical behavior can be discerned that combine an unusual mixture of second-order, infinite-order, and discontinuous transitions. In the spirit of Landau, the problem can be reduced to the local analysis of a cubic recursion equation, here, for the renormalization group flow of some generalized coupling. Among other insights, this theory explains the often noted prevalence of so-called inverted Berezinskii-Kosterlitz-Thouless transitions in complex networks. As a demonstration, a one-parameter family of Ising models on hierarchical networks is considered.
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http://arxiv.org/abs/1209.3447
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