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