A remark on Gibbs measures with log-correlated Gaussian fields

Abstract

We study Gibbs measures with log-correlated base Gaussian fields on the $d$-dimensional torus. In the defocusing case, the construction of such Gibbs measures follows from Nelson's argument. In this paper, we consider the focusing case with a quartic interaction. Using the variational formulation, we prove non-normalizability of the Gibbs measure. When $d = 2$, our argument provides an alternative proof of the non-normalizability result for the focusing $\Phi^4_2$-measure by Brydges and Slade (1996). Furthermore, we provide a precise rate of divergence, where the constant is characterized by the optimal constant for a certain Bernstein's inequality on $\mathbb{R}^d$. We also go over the construction of the focusing Gibbs measure with a cubic interaction. In the appendices, we present (a) non-normalizability of the Gibbs measure for the two-dimensional Zakharov system and (b) the construction of focusing quartic Gibbs measures with smoother base Gaussian measures, showing a critical nature of the log-correlated Gibbs measure with a focusing quartic interaction.