Origin of the growing length scale in M-p-spin glass models

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