  We develop the event-by-event picture of the gluon distribution produced via
medium-induced gluon branching by an energetic jet which propagates through a
dense QCD medium. A typical event is characterized by the production of a large
number of soft gluons which propagate at large angles with respect to the jet
axis and which collectively carry a substantial amount of energy. By explicitly
computing 2-gluon correlations, we demonstrate the existence of large
event-by-event fluctuations, which reflect the stochastic nature of the
branching process. For the two quantities that we have investigated -- the
energy loss at large angles and the soft gluon multiplicity --, the dispersion
is parametrically as large as the respective expectation value. We identify
interesting scaling laws, which suggest that the multiplicity distribution
should exhibit KNO (Koba-Nielsen-Olesen) scaling. A similar scaling is known to
hold for a jet branching in the vacuum, but the medium-induced distribution is
found to be considerably broader. We predict that event-by-event measurements
of the di-jet asymmetry in Pb+Pb collisions at the LHC should observe large
fluctuations in the number of soft hadrons propagating at large angles and also
in the total energy carried by these hadrons.
