Proof of off-diagonal long-range order in a mean-field trapped Bose gas via the Feynman--Kac formula

Abstract

We consider the non-interacting Bose gas of many bosons in dimension larger than two in a trap in a mean-field setting with a vanishing factor in front of the kinetic energy. The semi-classical setting is a particular case and was analysed in great detail in a special, interacting case in [DS21]. 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 the vanishing factor above a certain threshold and non-occurrence of ODLRO for 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 [Fe53]. For small values of the factor, we prove that all loops have the minimal length one, and for large ones we prove 100 percent condensation and identify the distribution of the long-loop lengths as the Poisson--Dirichlet distribution.