  As a guideline for experimental tests of the ideal glass transition (Random
Pinning Glass Transition, RPGT) that shall be induced in a system by randomly
pinning particles, we performed first-principle computations within the
Hypernetted chain approximation and numerical simulations of a Hard Sphere
model of glass-former. We obtain confirmation of the expected enhancement of
glassy behaviour under the procedure of random pinning, which consists in
freezing a fraction $c$ of randomly chosen particles in the positions they have
in an equilibrium configuration. We present the analytical phase diagram as a
function of $c$ and of the packing fraction $\phi$, showing a line of RPGT
ending in a critical point. We also obtain first microscopic results on
cooperative length-scales characterizing medium-range amorphous order in Hard
Spere glasses and indirect quantitative information on a key thermodynamic
quantity defined in proximity of ideal glass transitions, the amorphous surface
tension. Finally, we present numerical results of pair correlation functions
able to differentiate the liquid and the glass phases, as predicted by the
analytic computations.
