### Abstract

Let θ_{1}, . . . , θ_{n} be random variables from Dyson's circular β-ensemble with probability density function Const · ∏_{1≤j<k≤n} |e^{iθj} - e^{iθk} |^{β}. For each n ≥ 2 and β >0, we obtain some inequalities on E[p_{μ}(Z_{n})p_{ν}(Z_{n})], where Z_{n} = (e^{iθ1}, . . . , e^{iθn}) 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→∞}E[p_{μ}(Z_{n})p_{ν}(Z_{n})] = δ_{μν}( 2/β)^{l(μ)}z_{μ} for any β > 0 and partitions μ, ν lim_{m→∞} E[|p_{m}(Z_{n})|^{2}] = n for any β >0 and n = 2, 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 E[|p_{m}(Z_{n})|^{2}] for β = 1, 4. The central limit theorems of ∑^{n}_{j=1} g(e^{iθj}) 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.

Original language | English (US) |
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Pages (from-to) | 3279-3336 |

Number of pages | 58 |

Journal | Annals of Probability |

Volume | 43 |

Issue number | 6 |

DOIs | |

State | Published - Jan 1 2015 |

### Keywords

- Central limit theorem
- Circular beta-ensemble
- Haar-invariance
- Jack function
- Moment
- Partition
- Random matrix

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## Cite this

*Annals of Probability*,

*43*(6), 3279-3336. https://doi.org/10.1214/14-AOP960