  We analyze the spread of a localized peak of energy into vacuum for nonlinear
diffusive processes. In contrast with standard diffusion, the nonlinearity
results in a compact wave with a sharp front separating the perturbed region
from vacuum. In $d$ spatial dimensions, the front advances as $t^{1/(2+da)}$
according to hydrodynamics, with $a$ the nonlinearity exponent. We show that
fluctuations in the front position grow as $\sim t^{\mu} \eta$, where
$\mu<1/(2+da)$ is a new exponent that we measure and $\eta$ is a random
variable whose distribution we characterize. Fluctuating corrections to
hydrodynamic profiles give rise to an excess penetration into vacuum, revealing
scaling behaviors and robust features. We also examine the discharge of a
nonlinear rarefaction wave into vacuum. Our results suggest the existence of
universal scaling behaviors at the fluctuating level in nonlinear diffusion.
