A remark on Gibbs measures with log-correlated Gaussian fields
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| Format: | Preprint |
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2020
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| author | Oh, Tadahiro Seong, Kihoon Tolomeo, Leonardo |
| author_facet | Oh, Tadahiro Seong, Kihoon Tolomeo, Leonardo |
| contents | 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 $Φ^4_2$-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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2012_06729 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | A remark on Gibbs measures with log-correlated Gaussian fields Oh, Tadahiro Seong, Kihoon Tolomeo, Leonardo Probability Mathematical Physics Analysis of PDEs 60H30, 81T08, 35Q53, 35Q55, 35L71 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 $Φ^4_2$-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. |
| title | A remark on Gibbs measures with log-correlated Gaussian fields |
| topic | Probability Mathematical Physics Analysis of PDEs 60H30, 81T08, 35Q53, 35Q55, 35L71 |
| url | https://arxiv.org/abs/2012.06729 |