Fluctuations for the critical free Bose gas

TL;DR

This paper analyzes the critical free Bose gas in three dimensions, revealing how domain geometry influences critical exponents, loop formation, and fluctuation laws, with implications for understanding phase transitions.

Contribution

It provides new insights into the dependence of critical phenomena on domain geometry and boundary conditions, and derives non-Gaussian fluctuation laws for the Bose gas.

Findings

Critical exponents depend on domain geometry and boundary conditions.

Large loops are characterized by the Minakshisundaram-Pleijel expansion.

Fluctuations follow non-Gaussian laws governed by the regularized Fredholm determinant.

Abstract

We study the critical free Bose gas from a probabilistic vantage with a focus on the three-dimensional case. We obtain the critical exponents. These exponents and the occurrence of macroscopic loops subtly depend on the geometry of the domain and the boundary conditions, contrary to the subcritical and supercritical case. In particular, the second term of the Minakshisundaram-Pleijel expansion determines the emergence of large loops. We furthermore obtain non-Gaussian limit laws for the fluctuations, governed by the regularized Fredholm determinant of the Green operator.