## The Annals of Probability

### Moments of traces of circular beta-ensembles

#### Abstract

Let $\theta_{1},\ldots,\theta_{n}$ be random variables from Dyson’s circular $\beta$-ensemble with probability density function $\operatorname{Const}\cdot\prod_{1\leq j<k\leq n}|e^{i\theta_{j}}-e^{i\theta_{k}}|^{\beta}$. For each $n\geq2$ and $\beta>0$, we obtain some inequalities on $\mathbb{E}[p_{\mu}(Z_{n})\overline{p_{\nu}(Z_{n})}]$, where $Z_{n}=(e^{i\theta_{1}},\ldots,e^{i\theta_{n}})$ 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: $\lim_{n\to\infty}\mathbb{E}[p_{\mu}(Z_{n})\overline{p_{\nu}(Z_{n})}]=\delta_{\mu\nu}(\frac{2}{\beta})^{l(\mu)}z_{\mu}$ for any $\beta>0$ and partitions $\mu,\nu$; $\lim_{m\to\infty}\mathbb{E}[|p_{m}(Z_{n})|^{2}]=n$ for any $\beta>0$ and $n\geq2$, where $l(\mu)$ is the length of $\mu$ and $z_{\mu}$ is explicit on $\mu$. These results apply to the three important ensembles: COE ($\beta=1$), CUE ($\beta=2$) and CSE ($\beta=4$). We further examine the nonasymptotic behavior of $\mathbb{E}[|p_{m}(Z_{n})|^{2}]$ for $\beta=1,4$. The central limit theorems of $\sum_{j=1}^{n}g(e^{i\theta_{j}})$ 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.

#### Article information

Source
Ann. Probab., Volume 43, Number 6 (2015), 3279-3336.

Dates
Revised: August 2014
First available in Project Euclid: 11 December 2015

https://projecteuclid.org/euclid.aop/1449843631

Digital Object Identifier
doi:10.1214/14-AOP960

Mathematical Reviews number (MathSciNet)
MR3433582

Zentralblatt MATH identifier
06541358

#### Citation

Jiang, Tiefeng; Matsumoto, Sho. Moments of traces of circular beta-ensembles. Ann. Probab. 43 (2015), no. 6, 3279--3336. doi:10.1214/14-AOP960. https://projecteuclid.org/euclid.aop/1449843631

#### References

• [1] Blower, G. (2009). Random Matrices: High Dimensional Phenomena. London Mathematical Society Lecture Note Series 367. Cambridge Univ. Press, Cambridge.
• [2] Collins, B. (2003). Moments and cumulants of polynomial random variables on unitary groups, the Itzykson–Zuber integral, and free probability. Int. Math. Res. Not. IMRN 17 953–982.
• [3] Collins, B. and Matsumoto, S. (2009). On some properties of orthogonal Weingarten functions. J. Math. Phys. 50 113516, 14.
• [4] Diaconis, P. and Evans, S. N. (2001). Linear functionals of eigenvalues of random matrices. Trans. Amer. Math. Soc. 353 2615–2633.
• [5] Diaconis, P. and Shahshahani, M. (1994). On the eigenvalues of random matrices. J. Appl. Probab. 31A 49–62.
• [6] Dong, Z., Jiang, T. and Li, D. (2012). Circular law and arc law for truncation of random unitary matrix. J. Math. Phys. 53 013301, 14.
• [7] Durrett, R. (2010). Probability: Theory and Examples, 4th ed. Cambridge Univ. Press, Cambridge.
• [8] Dyson, F. J. (1962). Statistical theory of the energy levels of complex systems. I. J. Math. Phys. 3 140–156.
• [9] Dyson, F. J. (1962). Statistical theory of the energy levels of complex systems. II. J. Math. Phys. 3 166–175.
• [10] Dyson, F. J. (1962). Statistical theory of the energy levels of complex systems. III. J. Math. Phys. 3 1191–1198.
• [11] Forrester, P. J. (2010). Log-Gases and Random Matrices. London Mathematical Society Monographs Series 34. Princeton Univ. Press, Princeton, NJ.
• [12] Forrester, P. J. and Warnaar, S. O. (2008). The importance of the Selberg integral. Bull. Amer. Math. Soc. (N.S.) 45 489–534.
• [13] Hardy, G. H., Littlewood, J. E. and Pólya, G. (1988). Inequalities. Cambridge Univ. Press, Cambridge.
• [14] Jiang, T. (2006). How many entries of a typical orthogonal matrix can be approximated by independent normals? Ann. Probab. 34 1497–1529.
• [15] Jiang, T. (2009). A variance formula related to quantum conductance. Phys. Lett. A 373 2117–2121.
• [16] Jiang, T. (2009). The entries of circular orthogonal ensembles. J. Math. Phys. 50 063302, 13.
• [17] Jiang, T. (2010). The entries of Haar-invariant matrices from the classical compact groups. J. Theoret. Probab. 23 1227–1243.
• [18] Johansson, K. (1997). On random matrices from the compact classical groups. Ann. of Math. (2) 145 519–545.
• [19] Killip, R. (2008). Gaussian fluctuations for $\beta$ ensembles. Int. Math. Res. Not. IMRN 8 Art. ID rnn007, 19.
• [20] Macdonald, I. G. (1995). Symmetric Functions and Hall Polynomials, 2nd ed. The Clarendon Press, New York.
• [21] Matsumoto, S. and Novak, J. (2013). Jucys–Murphy elements and unitary matrix integrals. Int. Math. Res. Not. IMRN 2 362–397.
• [22] Mehta, M. L. (2004). Random Matrices, 3rd ed. Pure and Applied Mathematics (Amsterdam) 142. Elsevier/Academic Press, Amsterdam.
• [23] Pastur, L. and Vasilchuk, V. (2004). On the moments of traces of matrices of classical groups. Comm. Math. Phys. 252 149–166.
• [24] Rains, E. M. (1997). High powers of random elements of compact Lie groups. Probab. Theory Related Fields 107 219–241.
• [25] Stolz, M. (2005). On the Diaconis–Shahshahani method in random matrix theory. J. Algebraic Combin. 22 471–491.
• [26] Strauss, W. A. (1992). Partial Differential Equations: An Introduction. Wiley, New York.