  We numerically study the zero-temperature relaxation dynamics of several
glass-forming models to their inherent structures, following quenches from
equilibrium configurations sampled across a wide range of initial temperatures.
In a mean-field Mari-Kurchan model, we find that relaxation changes from a
power-law to an exponential decay below a well-defined temperature, consistent
with recent findings in mean-field $p$-spin models. By contrast, for
finite-dimensional systems, the relaxation is always algebraic, with a
non-trivial universal exponent at high temperatures crossing over to a harmonic
value at low temperatures. We demonstrate that this apparent evolution is
controlled by a temperature-dependent population of localised glassy
excitations. Our work unifies several recent lines of studies aiming at a
detailed characterisation of the complex potential energy landscape of
glass-formers, and challenges both mean-field and real space descriptions of
glasses.
