  In the semi-classical treatment, i.e. in a classical black hole geometry,
Hawking quanta emerge from trans-Planckian configurations because of scale
invariance. There is indeed no scale to stop the blue-shifting effect
encountered in the backward propagation towards the event horizon. On the
contrary, when taking into account the gravitational interactions neglected in
the semi-classical treatment, a UV scale stopping the blue-shift could be
dynamically engendered. To show that this is the case, we use a
non-perturbative treatment based on the large-N limit, where $N$ is the number
of matter fields. In this limit, the semi-classical treatment is the leading
contribution. Non-linear gravitational effects appear in the next orders and in
the first of these, the effects are governed by the two-point correlation
function of the energy-momentum tensor evaluated in the vacuum. Taking this
correlator into account, backward propagated modes are dissipated at a distance
from the horizon $\propto G\kappa$ when measured in a freely falling frame.
($G$ is Newton's constant and $\kappa$ the surface gravity.) This result can be
also obtained by considering light propagation in a stochastic ensemble of
metrics whose fluctuations are determined by the above correlator.
