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

Fixed frequency eigenfunction immersions and supremum norms of random waves

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

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.
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, (2007).

[2]

A. Ayache and N. Tzvetkov, $L^p$ properties of Gaussian random series,, Trans. Amer. Math. Soc., 360 (2008), 4425. doi: 10.1090/S0002-9947-08-04456-5.

[3]

P. Bartlett, Theoretical statistics,, lecture 14, ().

[4]

N. Burq and G. Lebeau, Injections de Sobolev probabilistes et applications,, Ann. Sci. Éc. Norm. Supér. (4), 46 (2013), 917.

[5]

Y. Canzani and B. Hanin, Scaling limit for the kernel of the spectral projector and remainder estimates in the Pointwise Weyl Law,, preprint, ().

[6]

J. Duistermaat and V. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics,, Invent. Math., 29 (1975), 39. 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. doi: 10.1016/j.jfa.2014.02.012.

[8]

L. Hörmander, The spectral function of an elliptic operator,, Acta Math., 121 (1968), 193. 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. 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. 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.

[12]

J.-M. Loubes and B. Pelletier, A kernel-based classifier on a Riemannian manifold,, Statist. Decisions, 26 (2008), 35. doi: 10.1524/stnd.2008.0911.

[13]

L. Nicolaescu, Complexity of random smooth functions on compact manifolds,, Indiana Univ. Math. J., 63 (2014), 1037. doi: 10.1512/iumj.2014.63.5321.

[14]

J. Neuheisel, The Asymptotic Distribution of Nodal Sets on Spheres,, Ph.D. Thesis, (2000).

[15]

B. Shiffman and S. Zelditch, Random polynomials of high degree and Levy concentration of measure,, Asian J. Math., 7 (2003), 627.

[16]

G. Tian, On a set of polarized Kähler metrics on algebraic manifolds,, J. Differential Geom., 32 (1990), 99.

[17]

N. Tzvetkov, Riemannian analog of a Paley-Zygmund theorem,, in Séminaire Équations aux dérivées partielles, (): 2008.

[18]

J. VanderKam, $L^\infty$ norms and quantum ergodicity on the sphere,, Internat. Math. Res. Notices, 7 (1997), 329. doi: 10.1155/S1073792897000238.

[19]

S. Zelditch, Real and complex zeros of Riemannian random waves,, in Spectral Analysis in Geometry and Number Theory, (2009), 321. doi: 10.1090/conm/484/09482.

[20]

S. Zelditch, Fine structure of Zoll spectra,, J. Func. Anal., 143 (1997), 415. doi: 10.1006/jfan.1996.2981.

[21]

S. Zelditch, Szegö kernels and a theorem of Tian,, Internat. Math Res. Notices, (1998), 317. doi: 10.1155/S107379289800021X.

show all references

References:
[1]

R. Adler and J. Taylor, Random Fields and Geometry,, Springer Monographs in Mathematics, (2007).

[2]

A. Ayache and N. Tzvetkov, $L^p$ properties of Gaussian random series,, Trans. Amer. Math. Soc., 360 (2008), 4425. doi: 10.1090/S0002-9947-08-04456-5.

[3]

P. Bartlett, Theoretical statistics,, lecture 14, ().

[4]

N. Burq and G. Lebeau, Injections de Sobolev probabilistes et applications,, Ann. Sci. Éc. Norm. Supér. (4), 46 (2013), 917.

[5]

Y. Canzani and B. Hanin, Scaling limit for the kernel of the spectral projector and remainder estimates in the Pointwise Weyl Law,, preprint, ().

[6]

J. Duistermaat and V. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics,, Invent. Math., 29 (1975), 39. 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. doi: 10.1016/j.jfa.2014.02.012.

[8]

L. Hörmander, The spectral function of an elliptic operator,, Acta Math., 121 (1968), 193. 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. 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. 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.

[12]

J.-M. Loubes and B. Pelletier, A kernel-based classifier on a Riemannian manifold,, Statist. Decisions, 26 (2008), 35. doi: 10.1524/stnd.2008.0911.

[13]

L. Nicolaescu, Complexity of random smooth functions on compact manifolds,, Indiana Univ. Math. J., 63 (2014), 1037. doi: 10.1512/iumj.2014.63.5321.

[14]

J. Neuheisel, The Asymptotic Distribution of Nodal Sets on Spheres,, Ph.D. Thesis, (2000).

[15]

B. Shiffman and S. Zelditch, Random polynomials of high degree and Levy concentration of measure,, Asian J. Math., 7 (2003), 627.

[16]

G. Tian, On a set of polarized Kähler metrics on algebraic manifolds,, J. Differential Geom., 32 (1990), 99.

[17]

N. Tzvetkov, Riemannian analog of a Paley-Zygmund theorem,, in Séminaire Équations aux dérivées partielles, (): 2008.

[18]

J. VanderKam, $L^\infty$ norms and quantum ergodicity on the sphere,, Internat. Math. Res. Notices, 7 (1997), 329. doi: 10.1155/S1073792897000238.

[19]

S. Zelditch, Real and complex zeros of Riemannian random waves,, in Spectral Analysis in Geometry and Number Theory, (2009), 321. doi: 10.1090/conm/484/09482.

[20]

S. Zelditch, Fine structure of Zoll spectra,, J. Func. Anal., 143 (1997), 415. doi: 10.1006/jfan.1996.2981.

[21]

S. Zelditch, Szegö kernels and a theorem of Tian,, Internat. Math Res. Notices, (1998), 317. doi: 10.1155/S107379289800021X.

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