  We derive the exact n-point current expectation values in the
Landauer-Buttiker non-equilibrium steady state of a multi terminal system with
star graph geometry and a point-like defect localised in the vertex. The
current cumulants are extracted from the connected correlation functions and
the cumulant generating function is established. We determine the moments, show
that the associated moment problem has a unique solution and reconstruct
explicitly the corresponding probability distribution. The basic building
blocks of this distribution are the probabilities of particle emission and
absorption from the heat reservoirs, driving the system away from equilibrium.
We derive and analyse in detail these probabilities, showing that they fully
describe the quantum transport problem in the system.
