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.