Moments of traces of circular beta-ensembles

Let .1,.,.n be random variables from Dyson.s circular .-ensemble with probability density function Const..1.j<k.n|ei.j.ei.k|.. For each n.2 and .>0, we obtain some inequalities on E[p.(Zn)p.(Zn)..............], where Zn=(ei.1,.,ei.n) and p. is the power-sum symmetric function for partition .. When .=2, our inequalities recover an identity by Diaconis and Evans for Haar-invariant unitary matrices. Further, we have the following: limn..E[p.(Zn)p.(Zn)..............]=...(2.)l(.)z. for any .>0 and partitions .,.; limm..E[|pm(Zn)|2]=n for any .>0 and n.2, where l(.) is the length of . and z. is explicit on .. These results apply to the three important ensembles: COE (.=1), CUE (.=2) and CSE (.=4). We further examine the nonasymptotic behavior of E[|pm(Zn)|2] for .=1,4. The central limit theorems of .nj=1g(ei.j) are obtained when (i) g(z) is a polynomial and .>0 is arbitrary, or (ii) g(z) has a Fourier expansion and .=1,4. The main tool is the Jack function.