## Electronic Journal of Probability

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

Jason Fulman

#### Abstract

Combining Stein's method with heat kernel techniques, we show that the trace of the $j$th power of an element of $U(n,\mathbb{C}), USp(n,\mathbb{C})$, or $SO(n,\mathbb{R})$ has a normal limit with error term $C \dot j/n$, 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.

#### Article information

Source
Electron. J. Probab., Volume 17 (2012), paper no. 66, 16 pp.

Dates
Accepted: 18 August 2012
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465062388

Digital Object Identifier
doi:10.1214/EJP.v17-2251

Mathematical Reviews number (MathSciNet)
MR2968673

Zentralblatt MATH identifier
1252.60012

Rights

#### Citation

Fulman, Jason. Stein's method, heat kernel, and traces of powers of elements of compact Lie groups. Electron. J. Probab. 17 (2012), paper no. 66, 16 pp. doi:10.1214/EJP.v17-2251. https://projecteuclid.org/euclid.ejp/1465062388

#### 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.
• Bump, Daniel; Gamburd, Alex. On the averages of characteristic polynomials from classical groups. Comm. Math. Phys. 265 (2006), no. 1, 227–274.
• Chatterjee, Sourav. Fluctuations of eigenvalues and second order Poincaré inequalities. Probab. Theory Related Fields 143 (2009), no. 1-2, 1–40.
• Collins, Benoït; Stolz, Michael. Borel theorems for random matrices from the classical compact symmetric spaces. Ann. Probab. 36 (2008), no. 3, 876–895.
• Diaconis, Persi; Evans, Steven N. Linear functionals of eigenvalues of random matrices. Trans. Amer. Math. Soc. 353 (2001), no. 7, 2615–2633.
• Diaconis, Persi; Shahshahani, Mehrdad. On the eigenvalues of random matrices. Studies in applied probability. J. Appl. Probab. 31A (1994), 49–62.
• 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.
• 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.
• 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.
• Durrett, Richard. Probability: theory and examples. Second edition. Duxbury Press, Belmont, CA, 1996. xiii+503 pp. ISBN: 0-534-24318-5
• Fulman, Jason. Stein's method and characters of compact Lie groups. Comm. Math. Phys. 288 (2009), no. 3, 1181–1201.
• 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
• 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.
• Johansson, Kurt. On random matrices from the compact classical groups. Ann. of Math. (2) 145 (1997), no. 3, 519–545.
• Jorgenson, Jay; Lang, Serge. The ubiquitous heat kernel. Mathematics unlimited-2001 and beyond, 655–683, Springer, Berlin, 2001.
• Lévy, Thierry. Schur-Weyl duality and the heat kernel measure on the unitary group. Adv. Math. 218 (2008), no. 2, 537–575.
• 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.
• Liu, Kefeng. Heat kernel and moduli space. Math. Res. Lett. 3 (1996), no. 6, 743–762.
• Liu, Kefeng. Heat kernel and moduli spaces. II. Math. Res. Lett. 4 (1997), no. 4, 569–588.
• 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.
• 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.
• Rains, E. M. Combinatorial properties of Brownian motion on the compact classical groups. J. Theoret. Probab. 10 (1997), no. 3, 659–679.
• Rains, E. M. High powers of random elements of compact Lie groups. Probab. Theory Related Fields 107 (1997), no. 2, 219–241.
• 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.
• Rinott, Yosef; Rotar, Vladimir. Normal approximations by Stein's method. Decis. Econ. Finance 23 (2000), no. 1, 15–29.
• 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.
• 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
• Saloff-Coste, L. Precise estimates on the rate at which certain diffusions tend to equilibrium. Math. Z. 217 (1994), no. 4, 641–677.
• Sarnak, Peter. Arithmetic quantum chaos. The Schur lectures (1992) (Tel Aviv), 183–236, Israel Math. Conf. Proc., 8, Bar-Ilan Univ., Ramat Gan, 1995.
• 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.
• 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.
• 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