The low-temperature thermal properties of dielectric crystals are governed by acoustic excitations with large wavelengths that are well described by plane waves. This is the Debye model, which rests on the assumption that the medium is an elastic continuum, holds true for acoustic wavelengths large on the microscopic scale fixed by the interatomic spacing, and gradually breaks down on approaching it. Glasses are characterized as well by universal low-temperature thermal properties, that are however anomalous with respect to those of the corresponding crystalline phases. Related universal anomalies also appear in the low-frequency vibrational density of states and, despite of a longstanding debate, still remain poorly understood. Using molecular dynamics simulations of a model monatomic glass of extremely large size, we show that in glasses the structural disorder undermines the Debye model in a subtle way: the elastic continuum approximation for the acoustic excitations breaks down abruptly on the mesoscopic, medium-range-order length-scale of about ten interatomic spacings, where it still works well for the corresponding crystalline systems. On this scale, the sound velocity shows a marked reduction with respect to the macroscopic value. This turns out to be closely related to the universal excess over the Debye model prediction found in glasses at frequencies of ~1 THz in the vibrational density of states or at temperatures of ~10 K in the specific heat.
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