  We consider the path-integral representation of the ideal Bose gas under
various boundary conditions. We show that Bose--Einstein condensation occurs at
the famous critical density threshold, by proving that its $1$-particle-reduced
density matrix exhibits off-diagonal long-range order above that threshold, but
not below. Our proofs are based on the well-known Feynman--Kac formula and a
representation in terms of a crucial Poisson point process. Furthermore, in the
condensation regime, we derive a law of large numbers with strong concentration
for the number of particles in short loops. In contrast to the situation for
free boundary conditions, where the entire condensate sits in just one loop,
for all other boundary conditions we obtain the limiting Poisson--Dirichlet
distribution for the collection of the lengths of all long loops. Our proofs
are new and purely probabilistic (apart from a standard eigenvalue expansion),
using elementary tools like Markov's inequality, Poisson point processes,
combinatorial formulas for cardinalities of particular partition sets, and
asymptotics for random walks with Pareto-distributed steps.
