On the rigidity of a hard-sphere glass near random close packing

We study theoretically and numerically the microscopic cause of the rigidity 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 ϕ c . This allows to define normal modes, and to apply recent results valid for elastic networks: rigidity is a non-local property of the packing geometry, and is characterized by some length scale l * which diverges at ϕ c (Wyart M., Nagel S. R. and Witten T. A., Europhys. Lett. , 72 (2005) 486; Wyart M., Silbert L. E., Nagel S. R. and Witten T. A., Phys. Rev. E , 72 (2005) 051306). We compute the scaling of the bulk and shear moduli near ϕ c , and speculate on the possible implications of these results for the glass transition.