2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/105515We 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 $ϕ_c$. 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 $ϕ_c$ [1, 2]. 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.7 pages, 4 figures. Figure 4 had a wrong unit in abscissa, which was correctedSoft Condensed MatterDisordered Systems and Neural NetworksStatistical Mechanicstext