Phase transition of singular Gibbs measures for three-dimensional Schr\"odinger-wave system

We study singular Gibbs measures for the three-dimensional Schr\"odinger-wave system, also known as the Yukawa system. Our primary result is the phase transition between weak and strong coupling cases, a phenomenon absent in one- and two-dimensional cases. Therefore, the three-dimensional model turns out to be critical, exhibiting the phase transition. In the weak coupling case, the Gibbs measure can be normalized as a probability measure and is shown to be singular with respect to the Gaussian free field. Conversely, in the strong coupling case, the Gibbs measure cannot be constructed as a probability measure. In particular, the finite-dimensional truncated Gibbs measures have no weak limit in an appropriate space, even up to a subsequence.