On a condition for type-I Bose–Einstein condensation in random potentials in d dimensions

Abstract In this paper we discuss Bose–Einstein condensation (BEC) in systems of pairwise non-interacting bosons in random potentials in d dimensions. Working in a rather general framework, we provide a “gap condition” which is sufficient to conclude existence of type-I BEC in probability and in the rth mean. We illustrate our results in the context of the well-known (one-dimensional) Luttinger–Sy model. Here, whenever the particle density exceeds a critical value, we show in addition that only the ground state is macroscopically occupied.