  We consider the non-interacting Bose gas of $N$ bosons in dimension $d\geq 3$
in a trap in a mean-field setting with a vanishing factor $a_N$ in front of the
kinetic energy. The choice $a_N=N^{-2/d}$ is the semi-classical setting and was
analysed in great detail in a special, interacting case in Deuchert and
Seiringer (2021). Using a version of the well-known Feynman--Kac representation
and a further representation in terms of a Poisson point process, we derive
precise asymptotics for the reduced one-particle density matrix, implying
off-diagonal long-range order (ODLRO, a well-known criterion for Bose--Einstein
condensation) for $a_N$ above a certain threshold and non-occurrence of ODLRO
for $a_N$ below that threshold. In particular, we relate the condensate and its
total mass to the amount of particles in long loops in the Feynman--Kac
formula, the order parameter that Feynman suggested in Feynman (1953). For
$a_N\ll N^{-2/d}$, we prove that all loops have the minimal length one, and for
$a_N\gg N^{-2/d}$ we prove 100 percent condensation and identify the
distribution of the long-loop lengths as the Poisson--Dirichlet distribution.
