# Secular Coefficients and the Holomorphic Multiplicative Chaos

@article{Najnudel2020SecularCA, title={Secular Coefficients and the Holomorphic Multiplicative Chaos}, author={Joseph Najnudel and Elliot Paquette and Nicholas J. Simm}, journal={arXiv: Probability}, year={2020} }

We study the secular coefficients of $N \times N$ random unitary matrices $U_{N}$ drawn from the Circular $\beta$-Ensemble, which are defined as the coefficients of $\{z^n\}$ in the characteristic polynomial $\det(1-zU_{N}^{*})$. When $\beta > 4$ we obtain a new class of limiting distributions that arise when both $n$ and $N$ tend to infinity simultaneously. We solve an open problem of Diaconis and Gamburd by showing that for $\beta=2$, the middle coefficient tends to zero as $N \to \infty$. We…

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