Phase transition of singular Gibbs measures for three-dimensional Schrödinger-wave system

We study the phase transition phenomenon of the singular Gibbs measure associated with the Schr¨odinger-wave systems, initiated by Lebowitz, Rose, and Speer (1988). In the three-dimensional case, this problem turns out to be critical, exhibiting a phase transition according to the size of the coupling constant. In the weakly coupling region, the Gibbs measure can be constructed as a probability measure, which is singular with respect to the Gaussian free field. On the other hand, in the strong coupling case, the Gibbs measure can not be normalized as a probability measure. In particular, the finite-dimensional truncated Gibbs measures have no weak limit, even up to a subsequence. The singularity of the Gibbs measure makes an additional difficulty in proving the non-convergence in the strong coupling case.