Dilute spin glasses at zero temperature and the ¹/ ₂-state Potts model

A random Ising model with random positive, negative and zero nearest-neighbour exchange coefficients (with probabilities p, q=p and r respectively) is considered at zero temperature. The s state Potts model in the limit s to ¹/₂ is shown to describe the statistics of those clusters of bonds which contain no frustrated plaquettes. The phase transition of this model is interpreted as corresponding to a 'Mattis spin-glass' ordering on these clusters, yielding an upper bound for the concentration p at which some spin-glass ordering must occur.