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  • Igor Wigman (a1) and Nadav Yesha (a2)


We study the fine-scale $L^{2}$ -mass distribution of toral Laplace eigenfunctions with respect to random position in two and three dimensions. In two dimensions, under certain flatness assumptions on the Fourier coefficients and generic restrictions on energy levels, both the asymptotic shape of the variance is determined and the limiting Gaussian law is established in the optimal Planck-scale regime. In three dimensions the asymptotic behaviour of the variance is analysed in a more restrictive scenario (“Bourgain’s eigenfunctions”). Other than the said precise results, lower and upper bounds are proved for the variance under more general flatness assumptions on the Fourier coefficients.



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Current address: Department of Mathematics, University of Haifa, 3498838 Haifa, Israel email



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1. Benatar, J. and Maffucci, R. W., Random waves on T3 : nodal area variance and lattice point correlations. Int. Math. Res. Not. IMRN 2017, doi:10.1093/imrn/rnx220.
2. Benatar, J., Marinucci, D. and Wigman, I., Planck-scale distribution of nodal length of arithmetic random waves. J. Anal. Math. (to appear).
3. Berry, M., Regular and irregular semiclassical wavefunctions. J. Phys. A 10(12) 1977, 20832091.
4. Berry, M., Semiclassical mechanics of regular and irregular motion. In Chaotic Behavior of Deterministic Systems (Les Houches, 1981), North-Holland (Amsterdam, 1983), 171271.
5. Bombieri, E. and Bourgain, J., A problem on sums of two squares. Int. Math. Res. Not. IMRN 2015(11) 2015, 33433407.
6. Bourgain, J., On toral eigenfunctions and the random wave model. Israel J. Math. 201(2) 2014, 611630.
7. Bourgain, J. and Rudnick, Z., On the geometry of the nodal lines of eigenfunctions of the two-dimensional torus. Ann. Henri Poincaré 12(6) 2011, 10271053.
8. Bourgain, J., Rudnick, Z. and Sarnak, P., Spatial statistics for lattice points on the sphere I: individual results. Bull. Iranian Math. Soc. 43(4) 2017, 361386; special issue in honor of Freydoon Shahidi’s 70th birthday.
9. Colin de Verdière, Y., Ergodicité et fonctions propres du Laplacien. Comm. Math. Phys. 102 1985, 497502.
10. de Courcy-Ireland, M., Small-scale equidistribution for random spherical harmonics. Preprint, 2017, arXiv:1711.01317.
11. Erdős, P. and Hall, R. R., On the angular distribution of Gaussian integers with fixed norm. Discrete Math. 200(1–3) 1999, 8794.
12. Feller, W., An Introduction to Probability Theory and its Applications, Vol. 2, second edn., Wiley (New York, 1971).
13. Gradhsteyn, I. S. and Rizhik, I. M., Tables of Integrals, Series and Products, 6th edn., Academic Press (2000).
14. Granville, A. and Wigman, I., Planck-scale mass equidistribution of toral Laplace eigenfunctions. Comm. Math. Phys. 355(2) 2017, 767802.
15. Han, X., Small scale quantum ergodicity in negatively curved manifolds. Nonlinearity 28(9) 2015, 32633288.
16. Han, X., Small scale equidistribution of random eigenbases. Comm. Math. Phys. 349(1) 2017, 425440.
17. Han, X. and Tacy, M., Equidistribution of random waves on small balls. Preprint, 2016, arXiv:1611.05983.
18. Hezari, H. and Rivière, G., Quantitative equidistribution properties of toral eigenfunctions. J. Spectr. Theory 7(2) 2017, 471485.
19. Hezari, H. and Rivière, G., L p norms, nodal sets, and quantum ergodicity. Adv. Math. 290 2016, 938966.
20. Humphries, P., Equidistribution in shrinking sets and L 4 -norm bounds for automorphic forms. Math. Ann. 371(3–4) 2018, 14971543.
21. Krishnapur, M., Kurlberg, P. and Wigman, I., Nodal length fluctuations for arithmetic random waves. Ann. of Math. (2) 177(2) 2013, 699737.
22. Kuipers, L. and Niederreiter, H., Uniform Distribution of Sequences, Wiley (New York, 1974).
23. Lester, S. and Rudnick, Z., Small scale equidistribution of eigenfunctions on the torus. Comm. Math. Phys. 350(1) 2017, 279300.
24. Luo, W. Z. and Sarnak, P., Quantum ergodicity of eigenfunctions on PSL2(ℤ) \backslashℍ2 . Publ. Math. Inst. Hautes Études Sci. 81 1995, 207237.
25. Sarnak, P., Variance sums on symmetric spaces. Private communication.
26. Sartori, A., Mass distribution for toral eigenfunctions via Bourgain’s de-randomisation. Preprint, 2018, arXiv:1812.00962.
27. Shnirelman, A., Ergodic properties of eigenfunctions. Uspekhi Mat. Nauk 180 1974, 181182.
28. Young, M., The quantum unique ergodicity conjecture for thin sets. Adv. Math. 286 2016, 9581016.
29. Zelditch, S., Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55 1987, 919941.
30. Zygmund, A., On Fourier coefficients and transforms of functions of two variables. Studia Math. 50 1974, 189201.
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  • Igor Wigman (a1) and Nadav Yesha (a2)


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