WIAS Preprint No. 3067, (2023)

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


  • König, Wolfgang
    ORCID: 0000-0002-7673-4364
  • Vogel, Quirin
  • Zass, Alexander
    ORCID: 0000-0001-6124-842X

2020 Mathematics Subject Classification

  • 60K35 82B10


  • Interacting many-particle systems, Bose--Einstein condensation, Brownian bridge, Feynman--Kac formula, long-range order, Poisson--Dirichlet distribution, partitions, random walks with heavy-tailed steps




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 (a part 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.

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