Limit theorems for statistics of combinatorial partitions with applications to mean field Bose gas

In this paper we study the statistics of combinatorial partitions of the integers, which arise when studying the occupation numbers of loops in the mean field Bose gas. We review the results of Lewis and collaborators and get some more precise estimates on the behavior at the critical point (fluctuations of the condensate component, finite volume corrections to the pressure). We then prove limit shape theorems for the loops occupation numbers. In particular we prove that in a certain range of the parameters, a finite fraction of the total mass is, in the limit, supported by infinitely long loops. We also show that this mass is equal to the mass of the condensed state where all particles have zero momentum.