The density of imaginary multiplicative chaos is positive
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866912732877422592 |
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| author | Aru, Juhan Jego, Antoine Junnila, Janne |
| author_facet | Aru, Juhan Jego, Antoine Junnila, Janne |
| contents | Consider a log-correlated Gaussian field $Γ$ and its associated imaginary multiplicative chaos $:e^{i βΓ}:$ where $β$ is a real parameter. In [AJJ22], we showed that for any nonzero test function $f$, the law of $\int f :e^{i βΓ}:$ possesses a smooth density with respect to Lebesgue measure on $\mathbb{C}$. In this note, we show that this density is strictly positive everywhere on $\mathbb{C}$. Our simple and direct strategy could be useful for studying other functionals on Gaussian spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_05289 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The density of imaginary multiplicative chaos is positive Aru, Juhan Jego, Antoine Junnila, Janne Probability 60G15, 60G20, 60G60 Consider a log-correlated Gaussian field $Γ$ and its associated imaginary multiplicative chaos $:e^{i βΓ}:$ where $β$ is a real parameter. In [AJJ22], we showed that for any nonzero test function $f$, the law of $\int f :e^{i βΓ}:$ possesses a smooth density with respect to Lebesgue measure on $\mathbb{C}$. In this note, we show that this density is strictly positive everywhere on $\mathbb{C}$. Our simple and direct strategy could be useful for studying other functionals on Gaussian spaces. |
| title | The density of imaginary multiplicative chaos is positive |
| topic | Probability 60G15, 60G20, 60G60 |
| url | https://arxiv.org/abs/2403.05289 |