Moments of traces of circular beta-ensembles. (English) Zbl 1388.60029

Jiang, Tiefeng; Matsumoto, Sho

Ann. Probab. 43, No. 6, 3279-3336 (2015).

Summary: Let \(\theta_1,\dots ,\theta_n\) be random variables from Dyson’s circular \(\beta\)-ensemble with probability density function \(\mathrm{Const}\cdot \prod_{1\leq j < k \leq n} |e^{i\theta_j} - e^{i\theta_k}|^\beta\). For each \(n\geq 2\) and \(\beta >0\), we obtain some inequalities on \(\mathbb E[p_\mu(Z_n)\overline {p_\nu (Z_n)}]\), where \(Z_n = (e^{i\theta_1},\dots, e^{i\theta_n})\) and \(p_\mu\) is the power-sum symmetric function for partition \(\mu\). When \(\beta = 2\), our inequalities recover an identity by P. Diaconis and S. N. Evans [Trans. Am. Math. Soc. 353, No. 7, 2615–2633 (2001; Zbl 1008.15013)] for Haar-invariant unitary matrices. Further, we have the following: \[ \lim_{n\to\infty} \mathbb E[p_\mu(Z_n)\overline{p_\nu(Z_n)}] = \delta_{\mu\nu}\left({\frac 2\beta}\right)^{l(\mu)}z_\mu \] 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 \(n\geq 2\), where \(l(\mu)\) is the length of \(\mu\) and \(z_\mu\) depends explicitly 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.

MSC:

60B20	Random matrices (probabilistic aspects)
15B52	Random matrices (algebraic aspects)
05E05	Symmetric functions and generalizations

Keywords:

random matrix; circular beta-ensemble; moment; Jack function; partition; Haar-invariance; central limit theorem

Citations:

Zbl 1008.15013

References:

[1]	Blower, G. (2009). Random Matrices : High Dimensional Phenomena. London Mathematical Society Lecture Note Series 367 . Cambridge Univ. Press, Cambridge. · Zbl 1278.60003 · doi:10.1017/CBO9781139107129
[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. · Zbl 1049.60091 · doi:10.1155/S107379280320917X
[3]	Collins, B. and Matsumoto, S. (2009). On some properties of orthogonal Weingarten functions. J. Math. Phys. 50 113516, 14. · Zbl 1304.22007 · doi:10.1063/1.3251304
[4]	Diaconis, P. and Evans, S. N. (2001). Linear functionals of eigenvalues of random matrices. Trans. Amer. Math. Soc. 353 2615-2633. · Zbl 1008.15013 · doi:10.1090/S0002-9947-01-02800-8
[5]	Diaconis, P. and Shahshahani, M. (1994). On the eigenvalues of random matrices. J. Appl. Probab. 31A 49-62. · Zbl 0807.15015 · doi:10.2307/3214948
[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. · Zbl 1273.15040 · doi:10.1063/1.3672885
[7]	Durrett, R. (2010). Probability : Theory and Examples , 4th ed. Cambridge Univ. Press, Cambridge. · Zbl 1202.60001 · doi:10.1017/CBO9780511779398
[8]	Dyson, F. J. (1962). Statistical theory of the energy levels of complex systems. I. J. Math. Phys. 3 140-156. · Zbl 0105.41604 · doi:10.1063/1.1703773
[9]	Dyson, F. J. (1962). Statistical theory of the energy levels of complex systems. II. J. Math. Phys. 3 166-175. · Zbl 0105.41604 · doi:10.1063/1.1703773
[10]	Dyson, F. J. (1962). Statistical theory of the energy levels of complex systems. III. J. Math. Phys. 3 1191-1198. · Zbl 0111.32703 · doi:10.1063/1.1703862
[11]	Forrester, P. J. (2010). Log-Gases and Random Matrices. London Mathematical Society Monographs Series 34 . Princeton Univ. Press, Princeton, NJ. · Zbl 1217.82003
[12]	Forrester, P. J. and Warnaar, S. O. (2008). The importance of the Selberg integral. Bull. Amer. Math. Soc. ( N.S. ) 45 489-534. · Zbl 1154.33002 · doi:10.1090/S0273-0979-08-01221-4
[13]	Hardy, G. H., Littlewood, J. E. and Pólya, G. (1988). Inequalities . Cambridge Univ. Press, Cambridge. · Zbl 0634.26008
[14]	Jiang, T. (2006). How many entries of a typical orthogonal matrix can be approximated by independent normals? Ann. Probab. 34 1497-1529. · Zbl 1107.15018 · doi:10.1214/009117906000000205
[15]	Jiang, T. (2009). A variance formula related to quantum conductance. Phys. Lett. A 373 2117-2121. · Zbl 1229.81121 · doi:10.1016/j.physleta.2009.04.035
[16]	Jiang, T. (2009). The entries of circular orthogonal ensembles. J. Math. Phys. 50 063302, 13. · Zbl 1216.60010 · doi:10.1063/1.3152217
[17]	Jiang, T. (2010). The entries of Haar-invariant matrices from the classical compact groups. J. Theoret. Probab. 23 1227-1243. · Zbl 1204.60010 · doi:10.1007/s10959-009-0241-7
[18]	Johansson, K. (1997). On random matrices from the compact classical groups. Ann. of Math. (2) 145 519-545. · Zbl 0105.41604 · doi:10.1063/1.1703773
[19]	Killip, R. (2008). Gaussian fluctuations for \(\beta\) ensembles. Int. Math. Res. Not. IMRN 8 Art. ID rnn007, 19. · Zbl 1205.82081 · doi:10.1093/imrn/rnn007
[20]	Macdonald, I. G. (1995). Symmetric Functions and Hall Polynomials , 2nd ed. The Clarendon Press, New York. · Zbl 0824.05059
[21]	Matsumoto, S. and Novak, J. (2013). Jucys-Murphy elements and unitary matrix integrals. Int. Math. Res. Not. IMRN 2 362-397. · Zbl 1312.05143 · doi:10.1093/imrn/rnr267
[22]	Mehta, M. L. (2004). Random Matrices , 3rd ed. Pure and Applied Mathematics ( Amsterdam ) 142 . Elsevier/Academic Press, Amsterdam. · Zbl 1107.15019
[23]	Pastur, L. and Vasilchuk, V. (2004). On the moments of traces of matrices of classical groups. Comm. Math. Phys. 252 149-166. · Zbl 1102.60001 · doi:10.1007/s00220-004-1231-3
[24]	Rains, E. M. (1997). High powers of random elements of compact Lie groups. Probab. Theory Related Fields 107 219-241. · Zbl 0868.60012 · doi:10.1007/s004400050084
[25]	Stolz, M. (2005). On the Diaconis-Shahshahani method in random matrix theory. J. Algebraic Combin. 22 471-491. · Zbl 1098.15020 · doi:10.1007/s10801-005-4629-x
[26]	Strauss, W. A. (1992). Partial Differential Equations : An Introduction . Wiley, New York. · Zbl 0817.35001