  Percolation in a scale-free hierarchical network is solved exactly by
renormalization-group theory, in terms of the different probabilities of
short-range and long-range bonds. A phase of critical percolation, with
algebraic (Berezinskii-Kosterlitz-Thouless) geometric order, occurs in the
phase diagram, in addition to the ordinary (compact) percolating phase and the
non-percolating phase. It is found that no connection exists between, on the
one hand, the onset of this geometric BKT behavior and, on the other hand, the
onsets of the highly clustered small-world character of the network and of the
thermal BKT transition of the Ising model on this network. Nevertheless, both
geometric and thermal BKT behaviors have inverted characters, occurring where
disorder is expected, namely at low bond probability and high temperature,
respectively. This may be a general property of long-range networks.

Disordered Systems and Neural Networks;Statistical Mechanics