We study interacting quantum Bose gases in thermal equilibrium on a lattice. In this framework, we establish convergence of the grand-canonical Gibbs states to their mean-field (classical field) and large-mass (classical particle) limit.

Our analysis is based onrepresentations in terms of ensembles of interacting random loops, namely the Ginibre loop ensemble for quantum bose gases and the Symanzik loop ensemble for classical scalar field theories. For small enough interactions, we obtain corresponding results in the infinite volume limit by means of cluster expansions. This is joint work with Juerg Froehlich, Antti Knowles, and Benjamin Schlein.

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