## Abstract

Let $\theta_{1},\ldots,\theta_{n}$ be random variables from Dyson’s circular $\beta$-ensemble with probability density function $\operatorname{Const}\cdot\prod_{1\leq j<k\leq n}|e^{i\theta_{j}}-e^{i\theta_{k}}|^{\beta}$. For each $n\geq2$ and $\beta>0$, we obtain some inequalities on $\mathbb{E}[p_{\mu}(Z_{n})\overline{p_{\nu}(Z_{n})}]$, where $Z_{n}=(e^{i\theta_{1}},\ldots,e^{i\theta_{n}})$ and $p_{\mu}$ is the power-sum symmetric function for partition $\mu$. When $\beta=2$, our inequalities recover an identity by Diaconis and Evans for Haar-invariant unitary matrices. Further, we have the following: $\lim_{n\to\infty}\mathbb{E}[p_{\mu}(Z_{n})\overline{p_{\nu}(Z_{n})}]=\delta_{\mu\nu}(\frac{2}{\beta})^{l(\mu)}z_{\mu}$ for any $\beta>0$ and partitions $\mu,\nu$; $\lim_{m\to\infty}\mathbb{E}[|p_{m}(Z_{n})|^{2}]=n$ for any $\beta>0$ and $n\geq2$, where $l(\mu)$ is the length of $\mu$ and $z_{\mu}$ is explicit on $\mu$. These results apply to the three important ensembles: COE ($\beta=1$), CUE ($\beta=2$) and CSE ($\beta=4$). We further examine the nonasymptotic behavior of $\mathbb{E}[|p_{m}(Z_{n})|^{2}]$ for $\beta=1,4$. The central limit theorems of $\sum_{j=1}^{n}g(e^{i\theta_{j}})$ are obtained when (i) $g(z)$ is a polynomial and $\beta>0$ is arbitrary, or (ii) $g(z)$ has a Fourier expansion and $\beta=1,4$. The main tool is the Jack function.

## Citation

Tiefeng Jiang. Sho Matsumoto. "Moments of traces of circular beta-ensembles." Ann. Probab. 43 (6) 3279 - 3336, November 2015. https://doi.org/10.1214/14-AOP960

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