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I will present a rigorous study of the perfect Bose-gas in the

presence of a homogeneous ergodic random potential. It is

demonstrated that the Lifshitz tail behaviour of the one-particle

spectrum reduces the critical dimensionality of the (generalized)

Bose-Einstein Condensation (BEC) to $d=1$. To tackle the

Off-Diagonal Long-Range Order (ODLRO) I will introduce the

space averaged one-body reduced density matrix. For a one

dimensional Poisson-type random potential we proved that

randomness enhances the exponential decay of this matrix in domain

free of the BEC.

These general results will then be applied to the Luttinger-Sy model in

which I can explicitely compute any of the physical quantities

(pressure, density, type of condensation, ODLRO...).