The Fredrickson-Andersen model with random pinning on Bethe lattices and its MCT transitions

We investigate the dynamics of the randomly pinned Fredrickson-Andersen model on the Bethe lattice. We find a line of random pinning dynamical transitions whose dynamical critical properties are in the same universality class of the $A_2$ and $A_3$ transitions of Mode Coupling Theory. The $A_3$ behavior appears at the terminal point, where the relaxation becomes logarithmic and the relaxation time diverges exponentially. We explain the critical behavior in terms of self-induced disorder and avalanches, strengthening the relationship discussed in recent works between glassy dynamics and Random Field Ising Model.