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Moments of traces of circular beta-ensembles

Let $θ_1,\ldots,θ_n$ be random variables from Dyson's circular $β$-ensemble with probability density function $\operatorname {Const}\cdot\prod_{1\leq j<k\leq n}|e^{iθ_j}-e^{iθ_k}|^β$. For each $n\geq2$ and $β>0$, we obtain some inequalities on $\mathbb{E}[p_μ(Z_n)\bar{p_ν(Z_n)}]$, where $Z_n=(e^{iθ_1},\ldots,e^{iθ_n})$ and $p_μ$ is the power-sum symmetric function for partition $μ$. When $β=2$, 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_μ(Z_n)\bar{p_ν(Z_n)}]= δ_{μν}(\frac{2}β)^{l(μ)}z_μ$ for any $β>0$ and partitions $μ,ν$; $\lim_{m\to\infty}\mathbb{E}[|p_m(Z_n)|^2]=n$ for any $β>0$ and $n\geq2$, where $l(μ)$ is the length of $μ$ and $z_μ$ is explicit on $μ$. These results apply to the three important ensembles: COE ($β=1$), CUE ($β=2$) and CSE ($β=4$). We further examine the nonasymptotic behavior of $\mathbb{E}[|p_m(Z_n)|^2]$ for $β=1,4$. The central limit theorems of $\sum_{j=1}^ng(e^{iθ_j})$ are obtained when (i) $g(z)$ is a polynomial and $β>0$ is arbitrary, or (ii) $g(z)$ has a Fourier expansion and $β=1,4$. The main tool is the Jack function.