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

Jason Fulman (University of Southern California)

Abstract


Combining Stein's method with heat kernel techniques, we show that the trace of the jth power of an element of U(n,C), USp(n,C), or SO(n,R) has a normal limit with error term C j/n, with C an absolute constant. In contrast to previous works, here j may be growing with n. The technique might prove useful in the study of the value distribution of approximate eigenfunctions of Laplacians.


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Pages: 1-16

Publication Date: August 18, 2012

DOI: 10.1214/EJP.v17-2251

References

  • Biane, Philippe. Free Brownian motion, free stochastic calculus and random matrices. Free probability theory (Waterloo, ON, 1995), 1--19, Fields Inst. Commun., 12, Amer. Math. Soc., Providence, RI, 1997. MR1426833
  • Bump, Daniel; Gamburd, Alex. On the averages of characteristic polynomials from classical groups. Comm. Math. Phys. 265 (2006), no. 1, 227--274. MR2217304
  • Chatterjee, Sourav. Fluctuations of eigenvalues and second order Poincaré inequalities. Probab. Theory Related Fields 143 (2009), no. 1-2, 1--40. MR2449121
  • Collins, Benoît; Stolz, Michael. Borel theorems for random matrices from the classical compact symmetric spaces. Ann. Probab. 36 (2008), no. 3, 876--895. MR2408577
  • Diaconis, Persi; Evans, Steven N. Linear functionals of eigenvalues of random matrices. Trans. Amer. Math. Soc. 353 (2001), no. 7, 2615--2633. MR1828463
  • Diaconis, Persi; Shahshahani, Mehrdad. On the eigenvalues of random matrices. Studies in applied probability. J. Appl. Probab. 31A (1994), 49--62. MR1274717
  • Döbler, Christian; Stolz, Michael. Stein's method and the multivariate CLT for traces of powers on the classical compact groups. Electron. J. Probab. 16 (2011), no. 86, 2375--2405. MR2861678
  • Döbler, C. and Stolz, M., Linear statistics of random matrix eigenvalues via Stein's method, arXiv:1205.5403 (2012).
  • Duits, Maurice; Johansson, Kurt. Powers of large random unitary matrices and Toeplitz determinants. Trans. Amer. Math. Soc. 362 (2010), no. 3, 1169--1187. MR2563725
  • Dumitriu, Ioana; Edelman, Alan. Global spectrum fluctuations for the $\beta$-Hermite and $\beta$-Laguerre ensembles via matrix models. J. Math. Phys. 47 (2006), no. 6, 063302, 36 pp. MR2239975
  • Durrett, Richard. Probability: theory and examples. Second edition. Duxbury Press, Belmont, CA, 1996. xiii+503 pp. ISBN: 0-534-24318-5 MR1609153
  • Fulman, Jason. Stein's method and characters of compact Lie groups. Comm. Math. Phys. 288 (2009), no. 3, 1181--1201. MR2504870
  • Fulman, J. and Röllin, A., Stein's method, heat kernel, and linear functions on the orthogonal groups, arXiv:1109.2975 (2011).
  • Grigor'yan, Alexander. Heat kernel and analysis on manifolds. AMS/IP Studies in Advanced Mathematics, 47. American Mathematical Society, Providence, RI; International Press, Boston, MA, 2009. xviii+482 pp. ISBN: 978-0-8218-4935-4 MR2569498
  • Hughes, C. P.; Rudnick, Z. Mock-Gaussian behaviour for linear statistics of classical compact groups. Random matrix theory. J. Phys. A 36 (2003), no. 12, 2919--2932. MR1986399
  • Johansson, Kurt. On random matrices from the compact classical groups. Ann. of Math. (2) 145 (1997), no. 3, 519--545. MR1454702
  • Jorgenson, Jay; Lang, Serge. The ubiquitous heat kernel. Mathematics unlimited—2001 and beyond, 655--683, Springer, Berlin, 2001. MR1852183
  • Lévy, Thierry. Schur-Weyl duality and the heat kernel measure on the unitary group. Adv. Math. 218 (2008), no. 2, 537--575. MR2407946
  • Liu, Kefeng. Heat kernels, symplectic geometry, moduli spaces and finite groups. Surveys in differential geometry: differential geometry inspired by string theory, 527--542, Surv. Differ. Geom., 5, Int. Press, Boston, MA, 1999. MR1772278
  • Liu, Kefeng. Heat kernel and moduli space. Math. Res. Lett. 3 (1996), no. 6, 743--762. MR1426532
  • Liu, Kefeng. Heat kernel and moduli spaces. II. Math. Res. Lett. 4 (1997), no. 4, 569--588. MR1470427
  • Maher, D., Brownian motion and heat kernels on compact Lie groups and symmetric spaces, Ph.D. thesis, University of New South Wales, 2006.
  • Meckes, Elizabeth. On the approximate normality of eigenfunctions of the Laplacian. Trans. Amer. Math. Soc. 361 (2009), no. 10, 5377--5399. MR2515815
  • Meckes, E., An infinitesimal version of Stein's method of exchangeable pairs, Stanford University Ph.D. thesis, 2006.
  • Pastur, L.; Vasilchuk, V. On the moments of traces of matrices of classical groups. Comm. Math. Phys. 252 (2004), no. 1-3, 149--166. MR2104877
  • Rains, E. M. Combinatorial properties of Brownian motion on the compact classical groups. J. Theoret. Probab. 10 (1997), no. 3, 659--679. MR1468398
  • Rains, E. M. High powers of random elements of compact Lie groups. Probab. Theory Related Fields 107 (1997), no. 2, 219--241. MR1431220
  • Reinert, Gesine. Couplings for normal approximations with Stein's method. Microsurveys in discrete probability (Princeton, NJ, 1997), 193--207, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 41, Amer. Math. Soc., Providence, RI, 1998. MR1630415
  • Rinott, Yosef; Rotar, Vladimir. Normal approximations by Stein's method. Decis. Econ. Finance 23 (2000), no. 1, 15--29. MR1780090
  • Rinott, Yosef; Rotar, Vladimir. 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), no. 4, 1080--1105. MR1484798
  • Rosenberg, Steven. The Laplacian on a Riemannian manifold. An introduction to analysis on manifolds. London Mathematical Society Student Texts, 31. Cambridge University Press, Cambridge, 1997. x+172 pp. ISBN: 0-521-46300-9; 0-521-46831-0 MR1462892
  • Saloff-Coste, L. Precise estimates on the rate at which certain diffusions tend to equilibrium. Math. Z. 217 (1994), no. 4, 641--677. MR1306030
  • Sarnak, Peter. Arithmetic quantum chaos. The Schur lectures (1992) (Tel Aviv), 183--236, Israel Math. Conf. Proc., 8, Bar-Ilan Univ., Ramat Gan, 1995. MR1321639
  • Sinai, Ya.; Soshnikov, A. Central limit theorem for traces of large random symmetric matrices with independent matrix elements. Bol. Soc. Brasil. Mat. (N.S.) 29 (1998), no. 1, 1--24. MR1620151
  • Soshnikov, Alexander. The central limit theorem for local linear statistics in classical compact groups and related combinatorial identities. Ann. Probab. 28 (2000), no. 3, 1353--1370. MR1797877
  • 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).
  • Stein, Charles. Approximate computation of expectations. Institute of Mathematical Statistics Lecture Notes—Monograph Series, 7. Institute of Mathematical Statistics, Hayward, CA, 1986. iv+164 pp. ISBN: 0-940600-08-0 MR0882007
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