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arXiv · 2502.14863

The Fourier coefficients of the holomorphic multiplicative chaos in the limit of large frequency

Joseph Najnudel·Elliot Paquette·Nick Simm·Truong Vu
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Abstract

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 $β> 2$.

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Joseph Najnudel, Elliot Paquette, Nick Simm, Truong Vu. 2025-02-20. The Fourier coefficients of the holomorphic multiplicative chaos in the limit of large frequency. https://arxiv.org/abs/2502.14863

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