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Two versions of the M-p-spin glass model have been studied with theMigdal-Kadanoff renormalization group approximation. The model with p=3 and M=3has at mean-field level the ideal glass transition at the Kauzmann temperatureand at lower temperatures still the Gardner transition to a state like that ofan Ising spin glass in a field. The model with p=3 and M=2 has only the Gardnertransition. In the dimensions studied, d=2,3 and 4, both models behave almostidentically, indicating that the growing correlation length as the temperatureis reduced in these models -- the analogue of the point-to-set length scale --is not due to the mechanism postulated in the random first order transitiontheory of glasses, but is more like that expected on the analogy of glasses tothe Ising spin glass in a field.

Origin of the Growing Length Scale in M-p-Spin Glass Models