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Renormalization group analysis of the M-p-spin glass model with p=3 and M=3

We study an M-p-spin spin glass model with p=3 and M=3 in three dimensions using the Migdal-Kadanoff renormalization group approximation (MKA). In this version of the p-spin model, there are three (M=3) Ising spins on each site. At mean-field level, this model is known to have two transitions; a dynamical transition and a thermodynamic one at a lower temperature. The dynamical transition is similar to the mode-coupling transition in glasses, while the thermodynamic transition possibly describes what happens at the Kauzmann temperature. We find that all the coupling constants in the model flow under the MKA to the high-temperature sink implying that the mean-field features disappear in three dimensions and that there is no transition in this model. The behavior of the coupling constant flow is qualitatively similar to that of the model with p=3 and M=2, for which only a single transition is predicted at the mean-field level. We conclude that for p-spin models in three dimensions, fluctuation effects completely remove all traces of their mean-field behavior.