Localization of Random Surfaces with Monotone Potentials and an FKG-Gaussian Correlation Inequality
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911798336159744 |
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| author | Sellke, Mark |
| author_facet | Sellke, Mark |
| contents | The seminal 1975 work of Brascamp-Lieb-Lebowitz initiated the rigorous study of Ginzberg-Landau random surface models. It was conjectured therein that fluctuations are localized on $\mathbb Z^d$ when $d\geq 3$ for very general potentials, matching the behavior of the Gaussian free field. We confirm this behavior for all even potentials $U:\mathbb R\to\mathbb R$ satisfying $U'(x)\geq \min(\varepsilon x,\frac{1+\varepsilon}{x})$ on $x\in \mathbb R^+$. Given correspondingly stronger growth conditions on $U$, we show power or stretched exponential tail bounds on all transient graphs, which determine the maximum field value up to constants in many cases. Further extensions include non-wired boundary conditions and iterated Laplacian analogs such as the membrane model. Our main tool is an FKG-based generalization of the Gaussian correlation inequality, which is used to dominate the finite-volume Gibbs measures by mixtures of centered Gaussian fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_18737 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Localization of Random Surfaces with Monotone Potentials and an FKG-Gaussian Correlation Inequality Sellke, Mark Probability Mathematical Physics The seminal 1975 work of Brascamp-Lieb-Lebowitz initiated the rigorous study of Ginzberg-Landau random surface models. It was conjectured therein that fluctuations are localized on $\mathbb Z^d$ when $d\geq 3$ for very general potentials, matching the behavior of the Gaussian free field. We confirm this behavior for all even potentials $U:\mathbb R\to\mathbb R$ satisfying $U'(x)\geq \min(\varepsilon x,\frac{1+\varepsilon}{x})$ on $x\in \mathbb R^+$. Given correspondingly stronger growth conditions on $U$, we show power or stretched exponential tail bounds on all transient graphs, which determine the maximum field value up to constants in many cases. Further extensions include non-wired boundary conditions and iterated Laplacian analogs such as the membrane model. Our main tool is an FKG-based generalization of the Gaussian correlation inequality, which is used to dominate the finite-volume Gibbs measures by mixtures of centered Gaussian fields. |
| title | Localization of Random Surfaces with Monotone Potentials and an FKG-Gaussian Correlation Inequality |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2402.18737 |