The glass crossover from mean-field Spin-Glasses to supercooled liquids

Stochastic-Beta-Relaxation provides a characterisation of the glass crossover in discontinuous Spin-Glasses and supercoooled liquid. Notably it can be derived through a rigorous computation from a dynamical Landau theory. In this paper, I will discuss the precise meaning of this connection in a language that does not require familiarity with statistical field theory. I will discuss finite-size corrections in mean-field Spin-Glass models and loop corrections in finite-dimensional models that are both described by the dynamical Landau theory considered. Then I will argue that the same Landau theory can be associated to supercooled liquid described by Mode-Coupling Theory invoking a physical principle of time-scale invariance.