Fluctuations for the critical free Bose gas

We study the critical free Bose gas from a probabilistic perspective with a focus on the three-dimensional case.

We show that, unlike in the subcritical and supercritical regime, the critical asymptotics depend on the second heat-trace coefficient $a_1$ and hence on the geometry and boundary conditions.

For $a_1<0$, long loops occur on the scale $L^2\log L$ and yield Poisson--Dirichlet statistics; for $a_1<0$, such loops are suppressed and the relevant fluctuations are produced by the short-loop bulk.

We also derive asymptotics for the partition function, the one-particle reduced density matrix, and loop statistics.

The fluctuation limits are governed either by a regularized Fredholm determinant of the Green operator or by Gaussian local central limit theorems, depending on the boundary regime.