Derivatives of Gaussian multiplicative chaos
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866914283702452224 |
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| author | Jego, Antoine |
| author_facet | Jego, Antoine |
| contents | Consider a logarithmically-correlated Gaussian field $X$ in $d$ dimensions. For all $γ\in (-\sqrt{2d},\sqrt{2d})$, we show that the derivatives $\frac{\partial^k}{\partialγ^k} :e^{γX_ε}:$ of the regularised Gaussian multiplicative chaos $:e^{γX_ε}:$ converge as $ε\to 0$. By deriving optimal bounds on their growth as $k\to\infty$, we control the power expansion of $:e^{γX_ε}:$ about each $γ\in(-\sqrt{2d},\sqrt{2d})$. This yields an alternative approach to complex Gaussian multiplicative chaos in the whole subcritical regime, based entirely on real-valued quantities.
One of our key technical contributions is to provide a truncated second moment approach to the uniform integrability of the derivatives of multiplicative chaos and its associated complex variant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_19614 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Derivatives of Gaussian multiplicative chaos Jego, Antoine Probability Mathematical Physics Consider a logarithmically-correlated Gaussian field $X$ in $d$ dimensions. For all $γ\in (-\sqrt{2d},\sqrt{2d})$, we show that the derivatives $\frac{\partial^k}{\partialγ^k} :e^{γX_ε}:$ of the regularised Gaussian multiplicative chaos $:e^{γX_ε}:$ converge as $ε\to 0$. By deriving optimal bounds on their growth as $k\to\infty$, we control the power expansion of $:e^{γX_ε}:$ about each $γ\in(-\sqrt{2d},\sqrt{2d})$. This yields an alternative approach to complex Gaussian multiplicative chaos in the whole subcritical regime, based entirely on real-valued quantities. One of our key technical contributions is to provide a truncated second moment approach to the uniform integrability of the derivatives of multiplicative chaos and its associated complex variant. |
| title | Derivatives of Gaussian multiplicative chaos |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2601.19614 |