  A model of the mean-field interacting boson gas trapped by a weak harmonic
potential is considered by the \textit{boson random point fields} methods. We
prove that in the Weak Harmonic Trap (WHT) limit there are two phases
distinguished by the boson condensation and by a different behaviour of the
local particle density. For chemical potentials less than a certain critical
value, the resulting Random Point Field (RPF) coincides with the usual boson
RPF, which corresponds to a non-interacting (ideal) boson gas. For the chemical
potentials greater than the critical value, the boson RPF describes a divergent
(local) density, which is due to \textit{localization} of the macroscopic
number of condensed particles. Notice that it is this kind of transition that
observed in experiments producing the Bose-Einstein Condensation in traps.
