  We review two recent results on the ground state properties of bosonic
systems trapped by a double well external potential. In the limit of large
population and large separation between the wells we prove that fluctuations in
the number of particles occupying each single-well low energy mode occur at a
reduced scale with respect to $ \sqrt{N}$, the latter being the typical
prediction of the central limit theorem. This signals the breakdown of the
independent and uncorrelated particles picture of standard Bose-Einstein
condensation and the emergence of an interaction-driven correlated phase in the
ground state.
