Noise-like analytic properties of imaginary chaos
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916567687626752 |
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| author | Aru, Juhan Baverez, Guillaume Jego, Antoine Junnila, Janne |
| author_facet | Aru, Juhan Baverez, Guillaume Jego, Antoine Junnila, Janne |
| contents | In this note we continue the study of imaginary multiplicative chaos $μ_β:= \exp(i βΓ)$, where $Γ$ is a two-dimensional continuum Gaussian free field. We concentrate here on the fine-scale analytic properties of $|μ_β(Q(x,r))|$ as $r \to 0$, where $Q(x,r)$ is a square of side-length $2r$ centred at $x$. More precisely, we prove monofractality of this process, a law of the iterated logarithm as $r \to 0$ and analyse its exceptional points, which have a close connection to fast points of Brownian motion. Some of the technical ideas developed to address these questions also help us pin down the exact Besov regularity of imaginary chaos, a question left open in [JSW20]. All the mentioned properties illustrate the noise-like behaviour of the imaginary chaos. We conclude by proving that the processes $x \mapsto |μ_β(Q(x,r))|^2$, when normalised additively and multiplicatively, converge as $r \to 0$ in law, but not in probability, to white noise; this suggests that all the information of the multiplicative chaos is contained in the angular parts of $μ_β(Q(x,r))$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_14942 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Noise-like analytic properties of imaginary chaos Aru, Juhan Baverez, Guillaume Jego, Antoine Junnila, Janne Probability Mathematical Physics 60G15, 60G20, 60G60 In this note we continue the study of imaginary multiplicative chaos $μ_β:= \exp(i βΓ)$, where $Γ$ is a two-dimensional continuum Gaussian free field. We concentrate here on the fine-scale analytic properties of $|μ_β(Q(x,r))|$ as $r \to 0$, where $Q(x,r)$ is a square of side-length $2r$ centred at $x$. More precisely, we prove monofractality of this process, a law of the iterated logarithm as $r \to 0$ and analyse its exceptional points, which have a close connection to fast points of Brownian motion. Some of the technical ideas developed to address these questions also help us pin down the exact Besov regularity of imaginary chaos, a question left open in [JSW20]. All the mentioned properties illustrate the noise-like behaviour of the imaginary chaos. We conclude by proving that the processes $x \mapsto |μ_β(Q(x,r))|^2$, when normalised additively and multiplicatively, converge as $r \to 0$ in law, but not in probability, to white noise; this suggests that all the information of the multiplicative chaos is contained in the angular parts of $μ_β(Q(x,r))$. |
| title | Noise-like analytic properties of imaginary chaos |
| topic | Probability Mathematical Physics 60G15, 60G20, 60G60 |
| url | https://arxiv.org/abs/2401.14942 |