From explosive to infinite-order transitions on a hyperbolic network

Abstract

We analyze the phase transitions that emerge from the recursive design of certain hyperbolic networks that includes, for instance, a discontinuous ("explosive") transition in ordinary percolation. To this end, we solve the $q$-state Potts model in the analytic continuation for non-integer $q$ with the real-space renormalization group. We find exact expressions for this one-parameter family of models that describe the dramatic transformation of the transition. In particular, this variation in $q$ shows that the discontinuous transition is generic in the regime $q<2$ that includes percolation. A continuous ferromagnetic transition is recovered in a singular manner only for the Ising model, $q=2$. For $q>2$ the transition immediately transforms into an infinitely smooth order parameter of the Berezinskii-Kosterlitz-Thouless (BKT) type.

6 pages, 5 figures, http://www.physics.emory.edu/faculty/boettcher/

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• Percolation on branching simplicial and cell complexes and its relation to interdependent percolation
• Renormalization group for link percolation on planar hyperbolic manifolds
• Classification of critical phenomena in hierarchical small-world networks
• Renormalization group theory of percolation on pseudo-fractal simplicial and cell complexes
• The Griffiths Phase on Hierarchical Modular Networks with Small-world Edges
• Evaluation of Spectral Zeta-Functions with the Renormalization Group
• Renormalization Group Solution of the Chutes&Ladder Model