MOMENTS OF TRACES OF CIRCULAR BETA-ENSEMBLES

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