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Glasses have a large excess of low-frequency vibrational modes in comparisonwith continuous elastic body, the so-called Boson Peak, which appears tocorrelate with several crucial properties of glasses, such as transport orfragility. I review recent results showing that the Boson Peak is a necessaryconsequence of the weak connectivity of the solid. I explain why in assembliesrepulsive spheres the boson peak shifts up to zero frequency as the pressure islowered toward the jamming threshold, and derive the corresponding exponent. Ishow how these ideas capture the main low-frequency features of the vibrationalspectrum of amorphous silica. These results extend arguments of Phillips on thepresence of floppy modes in under-constrained covalent networks to glasseswhere the covalent network is rigid, or when interactions are purely radial.

Rigidity-based approach to the boson peak in amorphous solids: from sphere packing to amorphous silica