Glassy Critical Points and Random Field Ising Model

We consider the critical properties of points of continuous glass transition as one can find in liquids in presence of constraints or in liquids in porous media. Through a one loop analysis we show that the critical Replica Field Theory describing these points can be mapped in the $\phi^4$-Random Field Ising Model. We confirm our analysis studying the finite size scaling of the $p$-spin model defined on sparse random graph, where a fraction of variables is frozen such that the phase transition is of a continuous kind.
Comment: The paper has been completely revised. A completely new part with simulations of a p-spin glass model on random graph has been included. An appendix with the Mathematica worksheet used in the calculation of the diagrams has also been added