  For a variety of quenched random spin systems on an Apollonian network,
including ferromagnetic and antiferromagnetic bond percolation and the Ising
spin glass, we find the persistence of ordered phases up to infinite
temperature over the entire range of disorder. We develop a
renormalization-group technique that yields highly detailed information,
including the exact distributions of local magnetizations and local spin-glass
order parameters, which turn out to exhibit, as function of temperature,
complex and distinctive tulip patterns.
