Interacting many-particle systems in the random Kac–Luttinger model and proof of Bose–Einstein condensation

Abstract

Following a model originally considered by Kac and Luttinger, we study interacting many-particle systems in a random background. The background consists of hard spherical obstacles with fixed radius and that are distributed via a Poisson point process with constant intensity on $R^d$, $2\le d\in N$. Interactions among the (bosonic) particles are described through repulsive pair potentials of mean-field type. As a main result, we prove (complete) Bose–Einstein condensation (BEC) in the thermodynamic limit and into the minimizer of a Hartree-type functional, in probability or with probability almost one depending on the strength of the interaction. As an important ingredient, we use very recent results obtained by Alain-Sol Sznitman regarding the spectral gap of the Dirichlet Laplacian in a Poissonian cloud of hard spherical obstacles in large boxes. To the best of our knowledge, our paper provides the first proof of BEC for systems of interacting particles in the Kac–Luttinger model, or in fact for some higher-dimensional interacting random continuum model.