  We study the decaying Burgers dynamics in $d$ dimensions for random Gaussian
initial conditions. We focus on power-law initial energy spectra, such that the
system shows a self-similar evolution. This is the case of interest for the
"adhesion model" in cosmology and a standard framework for "decaying Burgers
turbulence". We briefly describe how the system can be studied through
perturbative expansions at early time or large scale (quasi-linear regime).
Next, we develop a saddle-point method, based on spherical instantons, that
allows to obtain the asymptotic probability distributions $\cP(\eta_r)$ and
$\cP(\ctheta_r)$, of the density and velocity increment over spherical cells,
reached in the quasi-linear regime. Finally, we show how this approach can be
extended to take into account the formation of shocks and we derive the
rare-event tails of these probability distributions, at any finite time and
scale. This also gives the high-mass tail of the mass function of point-like
singularities (shocks in the one dimensional case).
