Moments of traces of circular beta-ensembles

Let θ₁, . . . , θ_n be random variables from Dyson's circular β-ensemble with probability density function Const · ∏_1≤j<k≤n |e^iθ_j - e^iθ_k |^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^iθ_n) 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^iθ_j) 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.