WIAS Preprint No. 3110, (2024)

Self-repellent Brownian bridges in an interacting Bose gas


Authors

  • Bolthausen, Erwin
  • König, Wolfgang
    ORCID: 0000-0002-7673-4364
  • Mukherjee, Chiranjib

2020 Mathematics Subject Classification

  • 60F10 60J65 82B10 81S40

Keywords

  • Self-repellent Brownian bridges, interacting Bose gas, Dirichlet boundary condition, condensation phase transition, random geometric permutation, random partition, large deviations, lace expansion

DOI

10.20347/WIAS.PREPRINT.3110

Abstract

We consider a model of d-dimensional interacting quantum Bose gas, expressed in terms of an ensemble of interacting Brownian bridges in a large box and undergoing the influence of all the interactions between the legs of each of the Brownian bridges. We study the thermodynamic limit of the system and give an explicit formula for the limiting free energy and a necessary and sufficient criterion for the occurrence of a condensation phase transition. For d ≥ 5 and sufficiently small interaction, we prove that the condensate phase is not empty. The ideas of proof rely on the similarity of the interaction to that of the self-repellent random walk, and build on a lace expansion method conducive to treating paths undergoing mutual repellence within each bridge.

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