  We study the gluon distribution produced via successive medium-induced
branchings by an energetic jet propagating through a weakly-coupled quark-gluon
plasma. We show that under suitable approximations the evolution of the jet can
be described as a classical stochastic process, which is exactly solvable. For
this process, we construct exact analytic solutions for all the n-point
correlation functions (the n-body densities in the space of energy). The
corresponding results for the one-point and the two-point functions were
already known, but those for the higher-point functions are new. These results
demonstrate strong correlations associated with the existence of common
ancestors in the branching process. By integrating these n-point functions over
the gluon energies, we deduce the mean gluon multiplicity $\langle N\rangle$ as
well as the higher moments $\langle N^p\rangle$ with $p\ge 2$. We find that the
multiplicities of the soft gluons are parametrically large and show a
remarkable regularity, known as Koba-Nielsen-Olesen (KNO) scaling: the reduced
moments $\langle N^p\rangle/\langle N\rangle^p$ are pure numbers, independent
of any of the physical parameters of the problem. We recognize a special
negative binomial distribution which is characterized by large statistical
fluctuations. These predictions can be tested in Pb+Pb collisions at the LHC,
via event-by-event measurements of the di-jet asymmetry.
