Renormalization group for link percolation on planar hyperbolic manifolds.

Network geometry is currently a topic of growing scientific interest, as it opens the possibility to explore and interpret the interplay between structure and dynamics of complex networks using geometrical arguments. However, the field is still in its infancy. In this work we investigate the role of network geometry in determining the nature of the percolation transition in planar hyperbolic manifolds. Boettcher et al. [Nat. Comm. 3, 787 (2012)2041-172310.1038/ncomms1774] have shown that a special type of two-dimensional hyperbolic manifolds, the Farey graphs, display a discontinuous transition for ordinary link percolation. Here we investigate using the renormalization group the critical properties of link percolation on a wider class of two-dimensional hyperbolic deterministic and random manifolds constituting the skeletons of two-dimensional cell complexes. These hyperbolic manifolds are built iteratively by subsequently gluing m-polygons to single edges. We show that when the size m of the polygons is drawn from a distribution q_{m} with asymptotic power-law scaling q_{m}≃Cm^{-γ} for m≫1, different universality classes can be observed depending on the value of the power-law exponent γ. Interestingly, the percolation transition is hybrid for γ∈(3,4) and becomes continuous for γ∈(2,3].