Logarithmic critical slowing down in complex systems: from statics to dynamics

We consider second-order phase transitions in which the order parameter is a replicated overlap matrix. We focus on a tricritical point that occurs in a variety of mean-field models and that, more generically, describes higher order liquid-liquid or liquid-glass transitions. We show that the static replicated theory implies slowing down with a logarithmic decay in time. The dynamical equations turn out to be those predicted by schematic Mode Coupling Theory for supercooled viscous liquids at a $A_3$ singularity, where the parameter exponent is $\lambda =1$. We obtain a quantitative expression for the parameter $\mu$ of the logarithmic decay in terms of cumulants of the overlap, which are physically observable in experiments or numerical simulations.