2026/05/31 by Quirin Vogel
Mathematics · #math.PR #msc:60K35 #msc:82B10
32 pages, 2 figures, fixed typos of v1, comments welcome
arxiv created 2026/07/29 · arxiv updated 2026/07/30
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 a1 and hence on the geometry and boundary conditions. For a1<0, long loops occur on the scale L2log L and yield Poisson--Dirichlet statistics; for a1<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.