On the approximate normality of eigenfunctions of the Laplacian
Author:
Elizabeth Meckes
Journal:
Trans. Amer. Math. Soc. 361 (2009), 53775399
MSC (2000):
Primary 58J50; Secondary 60F05, 58J65
Published electronically:
May 4, 2009
MathSciNet review:
2515815
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: The main result of this paper is a bound on the distance between the distribution of an eigenfunction of the Laplacian on a compact Riemannian manifold and the Gaussian distribution. If is a random point on a manifold and is an eigenfunction of the Laplacian with norm one and eigenvalue , then This result is applied to construct specific examples of spherical harmonics of arbitrary (odd) degree which are close to Gaussian in distribution. A second application is given to random linear combinations of eigenfunctions on flat tori.
 1.
R.
Aurich and F.
Steiner, Statistical properties of highly excited quantum
eigenstates of a strongly chaotic system, Phys. D 64
(1993), no. 13, 185–214. MR 1214552
(94e:81058), http://dx.doi.org/10.1016/01672789(93)90255Y
 2.
M.
V. Berry, Regular and irregular semiclassical wavefunctions,
J. Phys. A 10 (1977), no. 12, 2083–2091. MR 0489542
(58 #8961)
 3.
Isaac
Chavel, Riemannian geometry—a modern introduction,
Cambridge Tracts in Mathematics, vol. 108, Cambridge University Press,
Cambridge, 1993. MR 1271141
(95j:53001)
 4.
Persi
Diaconis and David
Freedman, A dozen de Finettistyle results in search of a
theory, Ann. Inst. H. Poincaré Probab. Statist.
23 (1987), no. 2, suppl., 397–423 (English,
with French summary). MR 898502
(88f:60072)
 5.
Gerald
B. Folland, How to integrate a polynomial over a sphere, Amer.
Math. Monthly 108 (2001), no. 5, 446–448. MR
1837866, http://dx.doi.org/10.2307/2695802
 6.
Alfred
Gray, Tubes, 2nd ed., Progress in Mathematics, vol. 221,
Birkhäuser Verlag, Basel, 2004. With a preface by Vicente Miquel. MR 2024928
(2004j:53001)
 7.
Dennis
A. Hejhal and Barry
N. Rackner, On the topography of Maass waveforms for
𝑃𝑆𝐿(2,𝑍), Experiment. Math.
1 (1992), no. 4, 275–305. MR 1257286
(95f:11037)
 8.
G. Helleloid.
Personal communication.
 9.
Elizabeth
Meckes, Linear functions on the classical
matrix groups, Trans. Amer. Math. Soc.
360 (2008), no. 10, 5355–5366. MR 2415077
(2009f:60012), http://dx.doi.org/10.1090/S0002994708044449
 10.
Elizabeth
S. Meckes and Mark
W. Meckes, The central limit problem for random vectors with
symmetries, J. Theoret. Probab. 20 (2007),
no. 4, 697–720. MR 2359052
(2009b:60079), http://dx.doi.org/10.1007/s1095900701195
 11.
Peter
Sarnak, Arithmetic quantum chaos, The Schur lectures (1992)
(Tel Aviv), Israel Math. Conf. Proc., vol. 8, BarIlan Univ., Ramat
Gan, 1995, pp. 183–236. MR 1321639
(96d:11059)
 12.
C. Stein.
The accuracy of the normal approximation to the distribution of the traces of powers of random orthogonal matrices. Technical Report No. 470, Stanford University Department of Statistics, 1995.
 13.
N.
Ja. Vilenkin, Special functions and the theory of group
representations, Translated from the Russian by V. N. Singh.
Translations of Mathematical Monographs, Vol. 22, American Mathematical
Society, Providence, R. I., 1968. MR 0229863
(37 #5429)
 1.
 R. Aurich and F. Steiner.
Statistical properties of highly excited quantum eigenstates of a strongly chaotic system. Phys. D, 64(13):185214, 1993. MR 1214552 (94e:81058)
 2.
 M. V. Berry.
Regular and irregular semiclassical wavefunctions. J. Phys. A, 10(12):20832091, 1977. MR 0489542 (58:8961)
 3.
 I. Chavel.
Riemannian Geometrya Modern Introduction, volume 108 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1993. MR 1271141 (95j:53001)
 4.
 P. Diaconis and D. Freedman.
A dozen de Finettistyle results in search of a theory. Ann. Inst. H. Poincaré Probab. Statist., 23(2, suppl.):397423, 1987. MR 898502 (88f:60072)
 5.
 G. Folland.
How to integrate a polynomial over a sphere. Amer. Math. Monthly, 108(5):446448, 2001. MR 1837866
 6.
 Alfred Gray.
Tubes, volume 221 of Progress in Mathematics. Birkhäuser Verlag, Basel, second edition, 2004. With a preface by Vicente Miquel. MR 2024928 (2004j:53001)
 7.
 D. Hejhal and B. Rackner.
On the topography of Maass waveforms for . Experiment. Math., 1(4):275305, 1992. MR 1257286 (95f:11037)
 8.
 G. Helleloid.
Personal communication.
 9.
 E. Meckes.
Linear functions on the classical matrix groups. Trans. Amer. Math. Soc., 360(10):53555366, 2008. MR 2415077
 10.
 E. Meckes and M. Meckes.
The central limit problem for random vectors with symmetries. J. Theoret. Probab., 20(4):697720, 2007. MR 2359052
 11.
 P. Sarnak.
Arithmetic quantum chaos. In The Schur Lectures (1992) (Tel Aviv), volume 8 of Israel Math. Conf. Proc., pages 183236. BarIlan Univ., Ramat Gan, 1995. MR 1321639 (96d:11059)
 12.
 C. Stein.
The accuracy of the normal approximation to the distribution of the traces of powers of random orthogonal matrices. Technical Report No. 470, Stanford University Department of Statistics, 1995.
 13.
 N. Vilenkin.
Special Functions and the Theory of Group Representations. Translated from the Russian by V. N. Singh. Translations of Mathematical Monographs, Vol. 22. American Mathematical Society, Providence, R. I., 1968. MR 0229863 (37:5429)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2000):
58J50,
60F05,
58J65
Retrieve articles in all journals
with MSC (2000):
58J50,
60F05,
58J65
Additional Information
Elizabeth Meckes
Affiliation:
Department of Mathematics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, Ohio 44122
Email:
elizabeth.meckes@case.edu
DOI:
http://dx.doi.org/10.1090/S0002994709046613
PII:
S 00029947(09)046613
Keywords:
Eigenfunctions,
Laplacian,
value distributions,
spherical harmonics,
random waves,
Stein's method,
normal approximation
Received by editor(s):
May 17, 2007
Received by editor(s) in revised form:
October 9, 2007
Published electronically:
May 4, 2009
Additional Notes:
This research was supported by fellowships from the ARCS Foundation and the American Institute of Mathematics.
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
