American Institute of Mathematical Sciences

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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

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, (2007).   Google Scholar

[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.  Google Scholar

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

[5]

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

[6]

J. Duistermaat and V. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics,, Invent. Math., 29 (1975), 39.  doi: 10.1007/BF01405172.  Google Scholar

[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.  Google Scholar

[8]

L. Hörmander, The spectral function of an elliptic operator,, Acta Math., 121 (1968), 193.  doi: 10.1007/BF02391913.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[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.  Google Scholar

[20]

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

[21]

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

show all references

References:
[1]

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

[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.  Google Scholar

[3]

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

[4]

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

[5]

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

[6]

J. Duistermaat and V. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics,, Invent. Math., 29 (1975), 39.  doi: 10.1007/BF01405172.  Google Scholar

[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.  Google Scholar

[8]

L. Hörmander, The spectral function of an elliptic operator,, Acta Math., 121 (1968), 193.  doi: 10.1007/BF02391913.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[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.  Google Scholar

[20]

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

[21]

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

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