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

TL;DR

This paper investigates the phase transition of singular Gibbs measures for the three-dimensional Schrödinger-wave system, revealing a critical behavior absent in lower dimensions and highlighting the measure's singularity and non-existence in strong coupling.

Contribution

It demonstrates the existence of a phase transition in the three-dimensional model, showing the measure's singularity in weak coupling and non-constructibility in strong coupling, a novel insight for this system.

Findings

Phase transition between weak and strong coupling cases.

Gibbs measure is singular with respect to Gaussian free field in weak coupling.

Gibbs measure cannot be constructed as a probability measure in strong coupling.

Abstract

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.