  In a recent computer study, we have shown that the combination of spatially
heterogeneous dynamics and kinetic facilitation provides a microscopic
explanation for the emergence of excess wings in deeply supercooled liquids.
Motivated by these findings, we construct a minimal empirical model to describe
this physics and introduce dynamic facilitation in the trap model, which was
initially developed to capture the thermally-activated dynamics of glassy
systems. We fully characterise the relaxation dynamics of this facilitated trap
model varying the functional form of energy distributions and the strength of
dynamic facilitation, combining numerical results and analytic arguments.
Dynamic facilitation generically accelerates the relaxation of the deepest
traps, thus making relaxation spectra strongly asymmetric, with an apparent
"excess" signal at high frequencies. For well-chosen values of the parameters,
the obtained spectra mimic experimental results for organic liquids displaying
an excess wing. Overall, our results identify the minimal physical ingredients
needed to describe excess processes in relaxation spectra of supercooled
liquids.
