Two-step relaxation next to dynamic arrest in mean-field glasses: Spherical and Isingp-spin model

Critical slowing down dynamics of supercooled glass-forming liquids is usually understood at the mean-field level in the framework of mode coupling theory, providing a two-time relaxation scenario and power-law behaviors of the time correlation function at dynamic criticality. In this work we derive critical slowing down exponents of spin-glass models undergoing discontinuous transitions by computing their Gibbs free energy and connecting the dynamic behavior to static ``in-state'' properties. Both the spherical and Ising versions are considered and, in the simpler spherical case, a generalization to arbitrary schematic mode coupling kernels is presented. Comparison with dynamic results available in literature is performed. Analytical predictions for the Ising case are provided for any p.