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| Format: | Preprint |
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2025
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| Online-Zugang: | https://arxiv.org/abs/2502.14863 |
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| _version_ | 1866915163547893760 |
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| author | Najnudel, Joseph Paquette, Elliot Simm, Nick Vu, Truong |
| author_facet | Najnudel, Joseph Paquette, Elliot Simm, Nick Vu, Truong |
| contents | The holomorphic multiplicative chaos (HMC) is a holomorphic analogue of the Gaussian multiplicative chaos. It arises naturally as the limit in large matrix size of the characteristic polynomial of Haar unitary, and more generally circular-$β$-ensemble, random matrices.
We consider the Fourier coefficients of the holomorphic multiplicative chaos in the $L^1$-phase, and we show that appropriately normalized, this converges in distribution to a complex normal random variable, scaled by the total mass of the Gaussian multiplicative chaos measure on the unit circle. We further generalize this to a process convergence, showing the joint convergence of consecutive Fourier coefficients. As a corollary, we derive convergence in law of the secular coefficients of sublinear index of the circular-$β$-ensemble for all $β> 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_14863 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Fourier coefficients of the holomorphic multiplicative chaos in the limit of large frequency Najnudel, Joseph Paquette, Elliot Simm, Nick Vu, Truong Probability The holomorphic multiplicative chaos (HMC) is a holomorphic analogue of the Gaussian multiplicative chaos. It arises naturally as the limit in large matrix size of the characteristic polynomial of Haar unitary, and more generally circular-$β$-ensemble, random matrices. We consider the Fourier coefficients of the holomorphic multiplicative chaos in the $L^1$-phase, and we show that appropriately normalized, this converges in distribution to a complex normal random variable, scaled by the total mass of the Gaussian multiplicative chaos measure on the unit circle. We further generalize this to a process convergence, showing the joint convergence of consecutive Fourier coefficients. As a corollary, we derive convergence in law of the secular coefficients of sublinear index of the circular-$β$-ensemble for all $β> 2$. |
| title | The Fourier coefficients of the holomorphic multiplicative chaos in the limit of large frequency |
| topic | Probability |
| url | https://arxiv.org/abs/2502.14863 |