Moments of Traces for Circular Beta-ensembles

Let $\theta_1, \cdots, \theta_n$ be random variables from Dyson's circular $\beta$-ensemble with probability density function $Const\cdot \prod_{1\leq j< k\leq n}|e^{i\theta_j} - e^{i\theta_k}|^{\beta}.$ For each $n\geq 2$ and $\beta>0,$ we obtain some inequalities on $\mathbb{E} \big [p_{\mu}(Z_n)\overline{p_{\nu}(Z_n)}\,\big],$ where $Z_n=(e^{i\theta_1}, \cdots, e^{i\theta_n})$ 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 $$(a)\lim_{n\to\infty}\mathbb{E}\left[p_{\mu}(Z_n)\overline{p_{\nu}(Z_n)}\,\right] = \delta_{\mu \nu}\left(\frac{2}{\beta}\right)^{l(\mu)}z_{\mu} \quad \mbox{for any $\beta>0$ and partitions $\mu,\nu$};$$ $$(b) \lim_{m\to\infty}\mathbb{E}\left[ |p_m(Z_n)|^2 \right] = n \quad \mbox{for any $\beta>0$ and $n\geq 2$},$$ 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 ($\beta=4$). The main tool is the Jack function.