From explosive to infinite-order transitions on a hyperbolic network

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.