A remark on Gibbs measures with logcorrelated Gaussian fields
Abstract
We study Gibbs measures with logcorrelated 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 nonnormalizability of the Gibbs measure. When $d = 2$, our argument provides an alternative proof of the nonnormalizability 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) nonnormalizability of the Gibbs measure for the twodimensional Zakharov system and (b) the construction of focusing quartic Gibbs measures with smoother base Gaussian measures, showing a critical nature of the logcorrelated Gibbs measure with a focusing quartic interaction.
 Publication:

arXiv eprints
 Pub Date:
 December 2020
 DOI:
 10.48550/arXiv.2012.06729
 arXiv:
 arXiv:2012.06729
 Bibcode:
 2020arXiv201206729O
 Keywords:

 Mathematics  Probability;
 Mathematical Physics;
 Mathematics  Analysis of PDEs;
 60H30;
 81T08;
 35Q53;
 35Q55;
 35L71
 EPrint:
 38 pages. We added an argument, showing a precise rate of divergence