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

Infinitely Robust Order and Local Order-Parameter Tulips in Apollonian Networks with Quenched Disorder