  We present a rigorous study of the Bose-Einstein condensation in the
Luttinger-Sy model. We prove the existence of the condensation in this
one-dimensional model of the perfect boson gas placed in the Poisson random
potential of singular point impurities. To tackle the off-diagonal long-range
order we calculate explicitly the corresponding space-averaged one-body reduced
density matrix. We show that mathematical mechanism of the Bose-Einstein
condensation in this random model is similar to condensation in a
one-dimensional nonrandom hierarchical model of scaled intervals. For the
Luttinger-Sy model we prove the Kac-Luttinger conjecture, i.e., that this model
manifests a type I BEC localized in a single "largest" interval of logarithmic
size.
