Moments of traces of circular beta-ensembles

Abstract

Let ${\theta }_1,\dots ,{\theta }_n$ be random variables from Dyson's circular $\beta$-ensemble with probability density function $\mathrm{Const}\cdot {\prod }_{1\le j<k\le n}|e^{i{\theta }_j}-e^{i{\theta }_k}{|}^{\beta }$. For each $n\ge 2$ and $\beta >0$, we obtain some inequalities on $\mathbb{E}[p_{\mu }(Z_n)\overline{p_{\nu }(Z_n)}]$, where $Z_n=(e^{i{\theta }_1},\dots ,e^{i{\theta }_n})$ 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 $n\ge 2$, 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 ($\beta =4$). 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.

Published at http://dx.doi.org/10.1214/14-AOP960 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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