Construction of interacting thermal equilibrium states for fluctuations around a Bose-Einstein condensate and the infinite volum

In this talk we present the construction of equilibrium states at positive temperature with Bose-Einstein condensation, for a non-relativistic Bosonic QFT (gas of Bose particles) on $\mathbb{R}^3$, interacting through a localised two body interaction. We use methods of
quantum field theory in the algebraic formulation and of quantum statistical mechanics in the operator algebraic setting to obtain this result. Moreover, in order to prove convergence of the correlation functions, we rearrange the perturbative series introducing an auxiliary stochastic Gaussian field which mediates the interaction of the Bosonic field. Limits where the localisation of the two-body interaction is removed are eventually discussed in combination with other regimes. This talk is based on a collaboration with Nicola Pinamonti.