Gibbs measures with log-correlated Gaussian fields and its invariance under Hamiltonian PDEs

We study the Gibbs measure with log-correlated base Gaussian fields on the d-dimensional torus. When d = 2, the Gibbs measure corresponds to the well-studied Phi^k_2-measure. We first discuss the (non-)construction of the focusing Gibbs measure. As a Hamiltonian PDE system corresponding to the Gibbs measure, we consider the Zakharov-Yukawa system (Schr¨odinger-wave system with a Zakharov-type coupling) on the two-dimensional torus. We then present a phase transition and invariance of the Gibbs measure under the flow.

This is based on joint work with Tadahiro Oh (University of Edinburgh) and Leonardo Tolomeo (University of Bonn).