Geodesic
A remark on Gibbs measures with log-correlated Gaussian fields
Forum of Mathematics, Sigma, Tome 12 (2024)

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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 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 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. 
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     author = {Tadahiro Oh and Kihoon Seong and Leonardo Tolomeo},
     title = {A remark on {Gibbs} measures with log-correlated {Gaussian} fields},
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Tadahiro Oh; Kihoon Seong; Leonardo Tolomeo. A remark on Gibbs measures with log-correlated Gaussian fields. Forum of Mathematics, Sigma, Tome 12 (2024). doi: 10.1017/fms.2024.28

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