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We consider the path-integral representation of the ideal Bose gas undervarious boundary conditions. We show that Bose--Einstein condensation occurs atthe famous critical density threshold, by proving that its $1$-particle-reduceddensity matrix exhibits off-diagonal long-range order above that threshold, butnot below. Our proofs are based on the well-known Feynman--Kac formula and arepresentation in terms of a crucial Poisson point process. Furthermore, in thecondensation regime, we derive a law of large numbers with strong concentrationfor the number of particles in short loops. In contrast to the situation forfree boundary conditions, where the entire condensate sits in just one loop,for all other boundary conditions we obtain the limiting Poisson--Dirichletdistribution for the collection of the lengths of all long loops. Our proofsare 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, andasymptotics for random walks with Pareto-distributed steps.

Off-diagonal long-range order for the free Bose gas via the Feynman--Kac formula