Fixed frequency eigenfunction immersions and supremum norms of random waves

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2015, 22: 76-86. doi: 10.3934/era.2015.22.76

Fixed frequency eigenfunction immersions and supremum norms of random waves

Yaiza Canzani ^1, and Boris Hanin ^2,

1.	Department of Mathematics, Harvard University, Cambridge, United States
2.	Department of Mathematics, Massachusetts Institute of Technology, Cambridge, United States

Received January 2015 Revised June 2015 Published September 2015

Abstract
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Under a Creative Commons license

A compact Riemannian manifold may be immersed into Euclidean space by using high frequency Laplace eigenfunctions. We study the geometry of the manifold viewed as a metric space endowed with the distance function from the ambient Euclidean space. As an application we give a new proof of a result of Burq-Lebeau and others on upper bounds for the sup-norms of random linear combinations of high frequency eigenfunctions.

Keywords: Immersions by eigenfunctions, supremum norms, random waves..

Mathematics Subject Classification: 35P05 (Pri), 53C42 (Sec.

Citation: Yaiza Canzani, Boris Hanin. Fixed frequency eigenfunction immersions and supremum norms of random waves. Electronic Research Announcements, 2015, 22: 76-86. doi: 10.3934/era.2015.22.76

References:

[1]	R. Adler and J. Taylor, Random Fields and Geometry, Springer Monographs in Mathematics, Springer, New York, 2007.
[2]	A. Ayache and N. Tzvetkov, $L^p$ properties of Gaussian random series, Trans. Amer. Math. Soc., 360 (2008), 4425-4439. doi: 10.1090/S0002-9947-08-04456-5.
[3]	P. Bartlett, Theoretical statistics, lecture 14, Electronic notes. Available from: http://www.stat.berkeley.edu/~bartlett/courses/2013spring-stat210b/notes/14notes.pdf.
[4]	N. Burq and G. Lebeau, Injections de Sobolev probabilistes et applications, Ann. Sci. Éc. Norm. Supér. (4), 46 (2013), 917-962.
[5]	Y. Canzani and B. Hanin, Scaling limit for the kernel of the spectral projector and remainder estimates in the Pointwise Weyl Law, preprint, arXiv:1411.0658.
[6]	J. Duistermaat and V. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math., 29 (1975), 39-79. doi: 10.1007/BF01405172.
[7]	R. Feng and S. Zelditch, Median and mean of the supremum of $L^2$ normalized random holomorphic fields, J. Func. Anal., 266 (2014), 5085-5107. doi: 10.1016/j.jfa.2014.02.012.
[8]	L. Hörmander, The spectral function of an elliptic operator, Acta Math., 121 (1968), 193-218. doi: 10.1007/BF02391913.
[9]	M. Krishnapur, P. Kurlberg and I. Wigman, Nodal length fluctuations for arithmetic random waves, Ann. of Math. (2), 177 (2013), 699-737. doi: 10.4007/annals.2013.177.2.8.
[10]	F. Oravecz, Z. Rudnick and I. Wigman, The Leray measure of nodal sets for random eigenfunctions on the torus, Ann. Inst. Fourier (Grenoble), 58 (2008), 299-335. doi: 10.5802/aif.2351.
[11]	V. Ivriĭ, The second term of the spectral asymptotics for a Laplace-Beltrami operator on manifolds with boundary, (Russian) Funksional. Anal. i Prolzhen., 14 (1980), 25-34.
[12]	J.-M. Loubes and B. Pelletier, A kernel-based classifier on a Riemannian manifold, Statist. Decisions, 26 (2008), 35-51. doi: 10.1524/stnd.2008.0911.
[13]	L. Nicolaescu, Complexity of random smooth functions on compact manifolds, Indiana Univ. Math. J., 63 (2014), 1037-1065. doi: 10.1512/iumj.2014.63.5321.
[14]	J. Neuheisel, The Asymptotic Distribution of Nodal Sets on Spheres, Ph.D. Thesis, The Johns Hopkins University, 2000.
[15]	B. Shiffman and S. Zelditch, Random polynomials of high degree and Levy concentration of measure, Asian J. Math., 7 (2003), 627-646.
[16]	G. Tian, On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geom., 32 (1990), 99-130.
[17]	N. Tzvetkov, Riemannian analog of a Paley-Zygmund theorem, in Séminaire Équations aux dérivées partielles, 2008-2009, 1-14.
[18]	J. VanderKam, $L^\infty$ norms and quantum ergodicity on the sphere, Internat. Math. Res. Notices, 7 (1997), 329-347. doi: 10.1155/S1073792897000238.
[19]	S. Zelditch, Real and complex zeros of Riemannian random waves, in Spectral Analysis in Geometry and Number Theory, Contemp. Math., 484, Amer. Math. Soc., Providence, RI, 2009, 321-342. doi: 10.1090/conm/484/09482.
[20]	S. Zelditch, Fine structure of Zoll spectra, J. Func. Anal., 143 (1997), 415-460. doi: 10.1006/jfan.1996.2981.
[21]	S. Zelditch, Szegö kernels and a theorem of Tian, Internat. Math Res. Notices, (1998), 317-331. doi: 10.1155/S107379289800021X.

show all references

References:

[1]	R. Adler and J. Taylor, Random Fields and Geometry, Springer Monographs in Mathematics, Springer, New York, 2007.
[2]	A. Ayache and N. Tzvetkov, $L^p$ properties of Gaussian random series, Trans. Amer. Math. Soc., 360 (2008), 4425-4439. doi: 10.1090/S0002-9947-08-04456-5.
[3]	P. Bartlett, Theoretical statistics, lecture 14, Electronic notes. Available from: http://www.stat.berkeley.edu/~bartlett/courses/2013spring-stat210b/notes/14notes.pdf.
[4]	N. Burq and G. Lebeau, Injections de Sobolev probabilistes et applications, Ann. Sci. Éc. Norm. Supér. (4), 46 (2013), 917-962.
[5]	Y. Canzani and B. Hanin, Scaling limit for the kernel of the spectral projector and remainder estimates in the Pointwise Weyl Law, preprint, arXiv:1411.0658.
[6]	J. Duistermaat and V. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math., 29 (1975), 39-79. doi: 10.1007/BF01405172.
[7]	R. Feng and S. Zelditch, Median and mean of the supremum of $L^2$ normalized random holomorphic fields, J. Func. Anal., 266 (2014), 5085-5107. doi: 10.1016/j.jfa.2014.02.012.
[8]	L. Hörmander, The spectral function of an elliptic operator, Acta Math., 121 (1968), 193-218. doi: 10.1007/BF02391913.
[9]	M. Krishnapur, P. Kurlberg and I. Wigman, Nodal length fluctuations for arithmetic random waves, Ann. of Math. (2), 177 (2013), 699-737. doi: 10.4007/annals.2013.177.2.8.
[10]	F. Oravecz, Z. Rudnick and I. Wigman, The Leray measure of nodal sets for random eigenfunctions on the torus, Ann. Inst. Fourier (Grenoble), 58 (2008), 299-335. doi: 10.5802/aif.2351.
[11]	V. Ivriĭ, The second term of the spectral asymptotics for a Laplace-Beltrami operator on manifolds with boundary, (Russian) Funksional. Anal. i Prolzhen., 14 (1980), 25-34.
[12]	J.-M. Loubes and B. Pelletier, A kernel-based classifier on a Riemannian manifold, Statist. Decisions, 26 (2008), 35-51. doi: 10.1524/stnd.2008.0911.
[13]	L. Nicolaescu, Complexity of random smooth functions on compact manifolds, Indiana Univ. Math. J., 63 (2014), 1037-1065. doi: 10.1512/iumj.2014.63.5321.
[14]	J. Neuheisel, The Asymptotic Distribution of Nodal Sets on Spheres, Ph.D. Thesis, The Johns Hopkins University, 2000.
[15]	B. Shiffman and S. Zelditch, Random polynomials of high degree and Levy concentration of measure, Asian J. Math., 7 (2003), 627-646.
[16]	G. Tian, On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geom., 32 (1990), 99-130.
[17]	N. Tzvetkov, Riemannian analog of a Paley-Zygmund theorem, in Séminaire Équations aux dérivées partielles, 2008-2009, 1-14.
[18]	J. VanderKam, $L^\infty$ norms and quantum ergodicity on the sphere, Internat. Math. Res. Notices, 7 (1997), 329-347. doi: 10.1155/S1073792897000238.
[19]	S. Zelditch, Real and complex zeros of Riemannian random waves, in Spectral Analysis in Geometry and Number Theory, Contemp. Math., 484, Amer. Math. Soc., Providence, RI, 2009, 321-342. doi: 10.1090/conm/484/09482.
[20]	S. Zelditch, Fine structure of Zoll spectra, J. Func. Anal., 143 (1997), 415-460. doi: 10.1006/jfan.1996.2981.
[21]	S. Zelditch, Szegö kernels and a theorem of Tian, Internat. Math Res. Notices, (1998), 317-331. doi: 10.1155/S107379289800021X.

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