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 noninteger 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 type.