  We study theoretically and numerically the microscopic cause of the
mechanical stability of hard sphere glasses near their maximum packing. We show
that, after coarse-graining over time, the hard sphere interaction can be
described by an effective potential which is exactly logarithmic at the random
close packing $\phi_c$. This allows to define normal modes, and to apply recent
results valid for elastic networks: mechanical stability is a non-local
property of the packing geometry, and is characterized by some length scale
$l^*$ which diverges at $\phi_c$ [1, 2]. We compute the scaling of the bulk and
shear moduli near $\phi_c$, and speculate on the possible implications of these
results for the glass transition.
