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 phi(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 phi(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 phi(c), and speculate on the possible implications of these results for the glass transition