Stein's method, heat kernel, and traces of powers of elements of compact Lie groups

Electronic Journal of Probability

Volumn 17, Issue , 2012, Pages

  Fulman, Jason ^a  

^a UNIVERSITY OF SOUTHERN CALIFORNIA   (United States)

Author keywords

Heat kernel; Random matrix; Stein's method

Indexed keywords

EID: 84865823864     PISSN: None     EISSN: 10836489     Source Type: Journal    
DOI: 10.1214/EJP.v17-2251     Document Type: Article

References (37)

1. Biane, P., Free Brownian motion, free stochastic calculus and random matrices, in Free probability theory (Waterloo, ON, 1995), 1-19, Fields Inst. Commun., 12, Amer. Math. Soc., Providence, RI, 1997. MR-1426833
2. Bump, D. and Gamburd, A., On the averages of characteristic polynomials from classical groups, Comm. Math. Phys. 265 (2006), 227-274. MR-2217304
3. Chatterjee, S., Fluctuations of eigenvalues and second order Poincaré inequalities, Probab. Theory Related Fields 143 (2009), 1-40. MR-2449121
4. Collins, B. and Stolz, M., Borel theorems for random matrices from the classical compact symmetric spaces, Ann. Probab. 36 (2008), 876-895. MR-2408577
5. Diaconis, P. and Evans, S., Linear functionals of eigenvalues of random matrices, Trans. Amer. Math. Soc. 353 (2001), 2615-2633 (electronic). MR-1828463
6. Diaconis, P. and Shahshahani, M., On the eigenvalues of random matrices. Studies in applied probability. J. Appl. Probab. 31A (1994), 49-62. MR-1274717
7. Döbler, C. and Stolz, M., Stein's method and the multivariate CLT for traces of powers on the classical compact groups, Electron. J. Probab. 16 (2011), 2375-2405. MR-2861678
8. Döbler, C. and Stolz, M., Linear statistics of random matrix eigenvalues via Stein's method, arXiv:1205.5403 (2012).
9. Duits, M. and Johansson, K., Powers of large random unitary matrices and Toeplitz determinants, Trans. Amer. Math. Soc. 362 (2010), 1169-1187. MR-2563725
10. Dumitriu, I. and Edelman, A., Global spectrum fluctuations for the beta-Hermite and beta-Laguerre ensembles via matrix models, J. Math. Phys. 47 (2006) no. 6, 063302, 36 pp. MR-2239975
11. Durrett, R., Probability: theory and examples. Second edition. Duxbury Press, Belmont, CA, 1996. MR-1609153
12. Fulman, J., Stein's method and characters of compact Lie groups, Comm. Math. Phys. 288 (2009), 1181-1201. MR-2504870
13. Fulman, J. and Röllin, A., Stein's method, heat kernel, and linear functions on the orthogonal groups, arXiv:1109.2975 (2011).
14. Grigor'yan, A., Heat kernel and analysis on manifolds, AMS/IP Studies in Advanced Mathematics, 47. American Mathematical Society, Providence, RI; International Press, Boston, MA, 2009. MR-2569498
15. Hughes, C. P. and Rudnick, Z., Mock-Gaussian behaviour for linear statistics of classical compact groups. Random matrix theory. J. Phys. A 36 (2003), 2919-2932. MR-1986399
16. Johansson, K., On random matrices from the compact classical groups, Ann. of Math. 145 (1997), 519-545. MR-1454702
17. Jorgenson, J. and Lang, S., The ubiquitous heat kernel. Mathematics unlimited-2001 and beyond, 655-683, Springer, Berlin, 2001. MR-1852183
18. Levy, T., Schur-Weyl duality and the heat kernel measure on the unitary group, Adv. Math. 218 (2008), 537-575. MR-2407946
19. Liu, K., Heat kernels, symplectic geometry, moduli spaces and finite groups, in Surveys in differential geometry: differential geometry inspired by string theory, 527-542, Surv. Differ. Geom., 5, Int. Press, Boston, MA, 1999. MR-1772278
20. Liu, K., Heat kernel and moduli spaces, Math. Res. Lett. 3 (1996), 743-762. MR-1426532
21. Liu, K., Heat kernel and moduli spaces II, Math. Res. Lett. 4 (1997), 569-588. MR-1470427
22. Maher, D., Brownian motion and heat kernels on compact Lie groups and symmetric spaces, Ph.D. thesis, University of New South Wales, 2006.
23. Meckes, E., On the approximate normality of eigenfunctions of the Laplacian, Trans. Amer. Math. Soc. 361 (2009), 5377-5399. MR-2515815
24. Meckes, E., An infinitesimal version of Stein's method of exchangeable pairs, Stanford University Ph.D. thesis, 2006.
25. Pastur, L. and Vasilchuk, V., On the moments of traces of matrices of classical groups, Comm. Math. Phys. 252 (2004), 149-166. MR-2104877
26. Rains, E. M., Combinatorial properties of Brownian motion on the compact classical groups, J. Theoret. Probab. 10 (1997), 659-679. MR-1468398
27. Rains, E. M., High powers of random elements of compact Lie groups, Probab. Theory Related Fields 107 (1997), 219-241. MR-1431220
28. Reinert, G., Couplings for normal approximations with Stein's method, in Microsurveys in discrete probability (Princeton, NJ, 1997), 193-207, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 41, Amer. Math. Soc., Providence, RI, 1998. MR-1630415
29. Rinott, Y. and Rotar, V., Normal approximations by Stein's method, Decis. Econ. Finance 23 (2000), 15-29. MR-1780090
30. Rinott, Y. and Rotar, V., On coupling constructions and rates in the CLT for dependent summands with applications to the antivoter model and weighted U-statistics, Ann. Appl. Probab. 7 (1997), 1080-1105. MR-1484798
31. Rosenberg, S., The Laplacian on a Riemannian manifold. An introduction to analysis on manifolds. London Mathematical Society Student Texts, 31. Cambridge University Press, Cambridge, 1997. MR-1462892
32. Saloff-Coste, L., Precise estimates on the rate at which certain diffusions tend to equilibrium, Math. Z. 217 (1994), 641-677. MR-1306030
33. Sarnak, P., Arithmetic quantum chaos. The Schur lectures (1992) (Tel Aviv), Israel Math. Conf. Proc., 8, Bar-Ilan Univ., Ramat Gan, 1995. MR-1321639
34. Sinai, Y. and Soshnikov, A., Central limit theorem for traces of large random symmetric matrices with independent matrix elements, Bol. Soc. Brasil. Mat. (N.S.) 29 (1998), 1-24. MR-1620151
35. Soshnikov, A., The central limit theorem for local linear statistics in classical compact groups and related combinatorial identities, Ann. Probab. 28 (2000), 1353-1370. MR-1797877
36. Stein, C., The accuracy of the normal approximation to the distribution of the traces of powers of random orthogonal matrices. Stanford University Statistics Department technical report no. 470, (1995).
37. Stein, C., Approximate computation of expectations. Institute of Mathematical Statistics Lecture Notes-Monograph Series, 7. Institute of Mathematical Statistics, Hayward, CA, 1986. MR-0882007