Let θ1, . . . , θn be random variables from Dyson's circular β-ensemble with probability density function Const · ∏1≤j<k≤n |eiθj - eiθk |β. For each n ≥ 2 and β >0, we obtain some inequalities on E[pμ(Zn)pν(Zn)], where Zn = (eiθ1, . . . , eiθn) and pμ 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: limn→∞E[pμ(Zn)pν(Zn)] = δμν( 2/β)l(μ)zμ for any β > 0 and partitions μ, ν limm→∞ E[|pm(Zn)|2] = n for any β >0 and n = 2, where l(μ) is the length of μ and zμ 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 E[|pm(Zn)|2] for β = 1, 4. The central limit theorems of ∑nj=1 g(eiθj) 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.
Bibliographical notePublisher Copyright:
© Institute of Mathematical Statistics, 2015.
- Central limit theorem
- Circular beta-ensemble
- Jack function
- Random matrix