I shall discuss a certain mean-field limit for the positive temperature equilibria (Gibbs states) of the interacting Bose gas. This serves as a toy model for the understanding of the phase transition towards a Bose-Einstein condensate when the temperature is lowered. The limit model we rigorously derive is a classical field theory, based on a certain non-linear Gibbs measure. The latter is a natural invariant of the non-linear Schrödinger equation approximating the dynamics of the Bose gas. It is also the basic ingredient of the Euclidean approach to constructive quantum field theory, as well as the large-time asymptote for the stochastic non-linear heat equation. A difficulty is that this non-linear Gibbs measure lives on low regularity distributional spaces, so that the non-linearity has to be understood in a renormalized sense. I shall put emphasis on the control of the renormalization procedure at the level of the quantum many-body model.

Based on joint work with Mathieu Lewin and Phan Thành Nam