research

Moments of traces of circular beta-ensembles

Abstract

Let θ1,,θn\theta_1,\ldots,\theta_n be random variables from Dyson's circular β\beta-ensemble with probability density function Const1j<kneiθjeiθkβ\operatorname {Const}\cdot\prod_{1\leq j<k\leq n}|e^{i\theta_j}-e^{i\theta _k}|^{\beta}. For each n2n\geq2 and β>0\beta>0, we obtain some inequalities on E[pμ(Zn)pν(Zn)ˉ]\mathbb{E}[p_{\mu}(Z_n)\bar{p_{\nu}(Z_n)}], where Zn=(eiθ1,,eiθn)Z_n=(e^{i\theta_1},\ldots,e^{i\theta_n}) and pμp_{\mu} is the power-sum symmetric function for partition μ\mu. When β=2\beta=2, our inequalities recover an identity by Diaconis and Evans for Haar-invariant unitary matrices. Further, we have the following: limnE[pμ(Zn)pν(Zn)ˉ]=δμν(2β)l(μ)zμ \lim_{n\to\infty}\mathbb{E}[p_{\mu}(Z_n)\bar{p_{\nu}(Z_n)}]= \delta_{\mu\nu}(\frac{2}{\beta})^{l(\mu)}z_{\mu} for any β>0\beta>0 and partitions μ,ν\mu,\nu; limmE[pm(Zn)2]=n\lim_{m\to\infty}\mathbb{E}[|p_m(Z_n)|^2]=n for any β>0\beta>0 and n2n\geq2, where l(μ)l(\mu) is the length of μ\mu and zμz_{\mu} is explicit on μ\mu. These results apply to the three important ensembles: COE (β=1\beta=1), CUE (β=2\beta=2) and CSE (β=4\beta=4). We further examine the nonasymptotic behavior of E[pm(Zn)2]\mathbb{E}[|p_m(Z_n)|^2] for β=1,4\beta=1,4. The central limit theorems of j=1ng(eiθj)\sum_{j=1}^ng(e^{i\theta_j}) are obtained when (i) g(z)g(z) is a polynomial and β>0\beta>0 is arbitrary, or (ii) g(z)g(z) has a Fourier expansion and β=1,4\beta=1,4. The main tool is the Jack function.Comment: 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

    Similar works

    Full text

    thumbnail-image

    Available Versions