  We provide a rigorous derivation of nonlinear Gibbs measures in two and three
space dimensions, starting from many-body quantum systems in thermal
equilibrium. More precisely, we prove that the grand-canonical Gibbs state of a
large bosonic quantum system converges to the Gibbs measure of a nonlinear
Schr{\"o}dinger-type classical field theory, in terms of partition functions
and reduced density matrices. The Gibbs measure thus describes the behavior of
the infinite Bose gas at criticality, that is, close to the phase transition to
a Bose-Einstein condensate. The Gibbs measure is concentrated on singular
distributions and has to be appropriately renormalized, while the quantum
system is well defined without any renormalization. By tuning a single real
parameter (the chemical potential), we obtain a counter-term for the diverging
repulsive interactions which provides the desired Wick renormalization of the
limit classical theory. The proof relies on a new estimate on the entropy
relative to quasi-free states and a novel method to control quantum variances.
