A remark on Gibbs measures with log-correlated Gaussian fields
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...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Cambridge University Press
2024-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424000288/type/journal_article |
| _version_ | 1797219414474489856 |
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| author | Tadahiro Oh Kihoon Seong Leonardo Tolomeo |
| author_facet | Tadahiro Oh Kihoon Seong Leonardo Tolomeo |
| author_sort | Tadahiro Oh |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-24T12:33:16Z |
| format | Article |
| id | doaj.art-bc41e50476cb4d44b84c91b313bfd29b |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-04-24T12:33:16Z |
| publishDate | 2024-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-bc41e50476cb4d44b84c91b313bfd29b2024-04-08T02:09:16ZengCambridge University PressForum of Mathematics, Sigma2050-50942024-01-011210.1017/fms.2024.28A remark on Gibbs measures with log-correlated Gaussian fieldsTadahiro Oh0https://orcid.org/0000-0003-2313-1145Kihoon Seong1Leonardo Tolomeo2The University of Edinburgh, and The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, EH9 3FD, United KingdomDepartment of Mathematics, Cornell University, 310 Malott Hall, Ithaca, NY, 14853, USA; E-mail:The University of Edinburgh, and The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, EH9 3FD, United Kingdom Mathematical Institute, Hausdorff Center for Mathematics, Universität Bonn, Bonn, Germany; E-mail: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.https://www.cambridge.org/core/product/identifier/S2050509424000288/type/journal_article60H3081T0835Q5335Q5535L71 |
| spellingShingle | Tadahiro Oh Kihoon Seong Leonardo Tolomeo A remark on Gibbs measures with log-correlated Gaussian fields Forum of Mathematics, Sigma 60H30 81T08 35Q53 35Q55 35L71 |
| title | A remark on Gibbs measures with log-correlated Gaussian fields |
| title_full | A remark on Gibbs measures with log-correlated Gaussian fields |
| title_fullStr | A remark on Gibbs measures with log-correlated Gaussian fields |
| title_full_unstemmed | A remark on Gibbs measures with log-correlated Gaussian fields |
| title_short | A remark on Gibbs measures with log-correlated Gaussian fields |
| title_sort | remark on gibbs measures with log correlated gaussian fields |
| topic | 60H30 81T08 35Q53 35Q55 35L71 |
| url | https://www.cambridge.org/core/product/identifier/S2050509424000288/type/journal_article |
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