Direction distribution for nodal components of random band-limited functions on surfaces

Eswarathasan, Suresh; Wigman, Igor

doi:10.1090/tran/8153

Direction distribution for nodal components of random band-limited functions on surfaces
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by Suresh Eswarathasan and Igor Wigman PDF

Trans. Amer. Math. Soc. 373 (2020), 7383-7428 Request permission

Abstract:

Let $(\mathcal {M},g)$ be a smooth compact Riemannian surface with no boundary. Given a smooth vector field $V$ with finitely many zeros on $\mathcal {M}$, we study the distribution of the number of tangencies to $V$ of the nodal components of random band-limited functions. It is determined that in the high-energy limit, these obey a universal deterministic law, independent of the surface $\mathcal {M}$ and the vector field $V$, that is supported precisely on the even integers $2 \mathbb {Z}_{> 0}$.

References

Ralph Abraham and Joel Robbin, Transversal mappings and flows, W. A. Benjamin, Inc., New York-Amsterdam, 1967. An appendix by Al Kelley. MR 0240836
A. Auffinger, A. Lerario, and E. Lundberg, Topologies of random geometric complexes on Riemannian manifolds in the thermodynamic limit. arXiv preprint arXiv:1812.09224 (2018).
Jean-Marc Azaïs and Mario Wschebor, Level sets and extrema of random processes and fields, John Wiley & Sons, Inc., Hoboken, NJ, 2009. MR 2478201, DOI 10.1002/9780470434642
Dmitry Beliaev and Igor Wigman, Volume distribution of nodal domains of random band-limited functions, Probab. Theory Related Fields 172 (2018), no. 1-2, 453–492. MR 3851836, DOI 10.1007/s00440-017-0813-x
M. V. Berry, Regular and irregular semiclassical wavefunctions, J. Phys. A 10 (1977), no. 12, 2083–2091. MR 489542
B. Driver, Analysis Tools with Applications, Lecture Notes http://www.math.ucsd.edu/~bdriver/231-02-03/Lecture_Notes/PDE-Anal-Book/analpde1.pdf, Springer.
Jochen Brüning, Über Knoten von Eigenfunktionen des Laplace-Beltrami-Operators, Math. Z. 158 (1978), no. 1, 15–21 (German). MR 478247, DOI 10.1007/BF01214561
Jochen Brüning and Dieter Gromes, Über die Länge der Knotenlinien schwingender Membranen, Math. Z. 124 (1972), 79–82 (German). MR 287202, DOI 10.1007/BF01142586
Valentina Cammarota, Domenico Marinucci, and Igor Wigman, On the distribution of the critical values of random spherical harmonics, J. Geom. Anal. 26 (2016), no. 4, 3252–3324. MR 3544960, DOI 10.1007/s12220-015-9668-5
Yaiza Canzani and Boris Hanin, Scaling limit for the kernel of the spectral projector and remainder estimates in the pointwise Weyl law, Anal. PDE 8 (2015), no. 7, 1707–1731. MR 3399136, DOI 10.2140/apde.2015.8.1707
Yaiza Canzani and Boris Hanin, $C^\infty$ scaling asymptotics for the spectral projector of the Laplacian, J. Geom. Anal. 28 (2018), no. 1, 111–122. MR 3745851, DOI 10.1007/s12220-017-9812-5
Yaiza Canzani and Peter Sarnak, Topology and nesting of the zero set components of monochromatic random waves, Comm. Pure Appl. Math. 72 (2019), no. 2, 343–374. MR 3896023, DOI 10.1002/cpa.21795
Javier Cilleruelo, The distribution of the lattice points on circles, J. Number Theory 43 (1993), no. 2, 198–202. MR 1207499, DOI 10.1006/jnth.1993.1017
Nguyen Viet Dang and Gabriel Rivière, Equidistribution of the conormal cycle of random nodal sets, J. Eur. Math. Soc. (JEMS) 20 (2018), no. 12, 3017–3071. MR 3871498, DOI 10.4171/JEMS/828
Harold Donnelly and Charles Fefferman, Nodal sets of eigenfunctions on Riemannian manifolds, Invent. Math. 93 (1988), no. 1, 161–183. MR 943927, DOI 10.1007/BF01393691
Alberto Enciso and Daniel Peralta-Salas, Submanifolds that are level sets of solutions to a second-order elliptic PDE, Adv. Math. 249 (2013), 204–249. MR 3116571, DOI 10.1016/j.aim.2013.08.026
P. Erdős and R. R. Hall, On the angular distribution of Gaussian integers with fixed norm, Discrete Math. 200 (1999), no. 1-3, 87–94. Paul Erdős memorial collection. MR 1692282, DOI 10.1016/S0012-365X(98)00329-X
Suresh Eswarathasan, Tangent nodal sets for random spherical harmonics, Probabilistic methods in geometry, topology and spectral theory, Contemp. Math., vol. 739, Amer. Math. Soc., [Providence], RI, [2019] ©2019, pp. 17–43. MR 4033912, DOI 10.1090/conm/739/14892
Victor Guillemin and Alan Pollack, Differential topology, AMS Chelsea Publishing, Providence, RI, 2010. Reprint of the 1974 original. MR 2680546, DOI 10.1090/chel/370
Damien Gayet and Jean-Yves Welschinger, Betti numbers of random nodal sets of elliptic pseudo-differential operators, Asian J. Math. 21 (2017), no. 5, 811–839. MR 3767266, DOI 10.4310/AJM.2017.v21.n5.a2
Damien Gayet and Jean-Yves Welschinger, Lower estimates for the expected Betti numbers of random real hypersurfaces, J. Lond. Math. Soc. (2) 90 (2014), no. 1, 105–120. MR 3245138, DOI 10.1112/jlms/jdu018
Damien Gayet and Jean-Yves Welschinger, Universal components of random nodal sets, Comm. Math. Phys. 347 (2016), no. 3, 777–797. MR 3551254, DOI 10.1007/s00220-016-2595-x
Lars Hörmander, The spectral function of an elliptic operator, Acta Math. 121 (1968), 193–218. MR 609014, DOI 10.1007/BF02391913
I. Kátai and I. Környei, On the distribution of lattice points on circles, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 19 (1976), 87–91 (1977). MR 506102
Manjunath Krishnapur, Pär Kurlberg, and Igor Wigman, Nodal length fluctuations for arithmetic random waves, Ann. of Math. (2) 177 (2013), no. 2, 699–737. MR 3010810, DOI 10.4007/annals.2013.177.2.8
Pär Kurlberg and Igor Wigman, On probability measures arising from lattice points on circles, Math. Ann. 367 (2017), no. 3-4, 1057–1098. MR 3623219, DOI 10.1007/s00208-016-1411-4
Pär Kurlberg and Igor Wigman, Variation of the Nazarov-Sodin constant for random plane waves and arithmetic random waves, Adv. Math. 330 (2018), 516–552. MR 3787552, DOI 10.1016/j.aim.2018.03.026
Peter D. Lax, Asymptotic solutions of oscillatory initial value problems, Duke Math. J. 24 (1957), 627–646. MR 97628
Alexander Logunov and Eugenia Malinnikova, Nodal sets of Laplace eigenfunctions: estimates of the Hausdorff measure in dimensions two and three, 50 years with Hardy spaces, Oper. Theory Adv. Appl., vol. 261, Birkhäuser/Springer, Cham, 2018, pp. 333–344. MR 3792104
Alexander Logunov, Nodal sets of Laplace eigenfunctions: proof of Nadirashvili’s conjecture and of the lower bound in Yau’s conjecture, Ann. of Math. (2) 187 (2018), no. 1, 241–262. MR 3739232, DOI 10.4007/annals.2018.187.1.5
Alexander Logunov, Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure, Ann. of Math. (2) 187 (2018), no. 1, 221–239. MR 3739231, DOI 10.4007/annals.2018.187.1.4
Fedor Nazarov and Mikhail Sodin, On the number of nodal domains of random spherical harmonics, Amer. J. Math. 131 (2009), no. 5, 1337–1357. MR 2555843, DOI 10.1353/ajm.0.0070
F. Nazarov and M. Sodin, Asymptotic laws for the spatial distribution and the number of connected components of zero sets of Gaussian random functions, Zh. Mat. Fiz. Anal. Geom. 12 (2016), no. 3, 205–278. MR 3522141, DOI 10.15407/mag12.03.205
Z. Rudnick and I. Wigman, Points on nodal lines with given direction, J. Spec. Thr., to appear (2019), arXiv preprint arXiv:1802.09603
Yu. G. Safarov, Asymptotics of a spectral function of a positive elliptic operator without a nontrapping condition, Funktsional. Anal. i Prilozhen. 22 (1988), no. 3, 53–65, 96 (Russian); English transl., Funct. Anal. Appl. 22 (1988), no. 3, 213–223 (1989). MR 961761, DOI 10.1007/BF01077627
Yu. Safarov and D. Vassiliev, The asymptotic distribution of eigenvalues of partial differential operators, Translations of Mathematical Monographs, vol. 155, American Mathematical Society, Providence, RI, 1997. Translated from the Russian manuscript by the authors. MR 1414899, DOI 10.1090/mmono/155
Peter Sarnak and Igor Wigman, Topologies of nodal sets of random band-limited functions, Comm. Pure Appl. Math. 72 (2019), no. 2, 275–342. MR 3896022, DOI 10.1002/cpa.21794
Mikhail Sodin, Lectures on random nodal portraits, Probability and statistical physics in St. Petersburg, Proc. Sympos. Pure Math., vol. 91, Amer. Math. Soc., Providence, RI, 2016, pp. 395–422. MR 3526834, DOI 10.15407/mag12.03.205
Peter Swerling, Statistical properties of the contours of random surfaces, IRE Trans. IT-8 (1962), 315–321. MR 0175653, DOI 10.1109/tit.1962.1057724
Gabor Szegö, Orthogonal Polynomials, American Mathematical Society Colloquium Publications, Vol. 23, American Mathematical Society, New York, 1939. MR 0000077
I. Wigman, On the expected Betti numbers of nodal sets for random fields, Analysis and PDE, to appear. arXiv:1903.00538 (2019).
Shing Tung Yau, Survey on partial differential equations in differential geometry, Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 3–71. MR 645729
Shing-Tung Yau, Open problems in geometry, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990) Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 1–28. MR 1216573