Electronic Journal of Probability

Phase singularities in complex arithmetic random waves

Federico Dalmao, Ivan Nourdin, Giovanni Peccati, and Maurizia Rossi

Full-text: Open access

Abstract

Complex arithmetic random waves are stationary Gaussian complex-valued solutions of the Helmholtz equation on the two-dimensional flat torus. We use Wiener-Itô chaotic expansions in order to derive a complete characterization of the second order high-energy behaviour of the total number of phase singularities of these functions. Our main result is that, while such random quantities verify a universal law of large numbers, they also exhibit non-universal and non-central second order fluctuations that are dictated by the arithmetic nature of the underlying spectral measures. Such fluctuations are qualitatively consistent with the cancellation phenomena predicted by Berry (2002) in the case of complex random waves on compact planar domains. Our results extend to the complex setting recent pathbreaking findings by Rudnick and Wigman (2008), Krishnapur, Kurlberg and Wigman (2013) and Marinucci, Peccati, Rossi and Wigman (2016). The exact asymptotic characterization of the variance is based on a fine analysis of the Kac-Rice kernel around the origin, as well as on a novel use of combinatorial moment formulae for controlling long-range weak correlations. As a by-product of our analysis, we also deduce explicit bounds in smooth distances for the second order non-central results evoked above.

Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 71, 45 pp.

Dates
Received: 26 November 2018
Accepted: 15 May 2019
First available in Project Euclid: 28 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1561687604

Digital Object Identifier
doi:10.1214/19-EJP321

Subjects
Primary: 60G60: Random fields 60B10: Convergence of probability measures 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 58J50: Spectral problems; spectral geometry; scattering theory [See also 35Pxx] 35P20: Asymptotic distribution of eigenvalues and eigenfunctions

Keywords
Berry’s cancellation complex arithmetic random waves high-energy limit limit theorems Laplacian nodal intersections phase singularities Wiener chaos

Rights
Creative Commons Attribution 4.0 International License.

Citation

Dalmao, Federico; Nourdin, Ivan; Peccati, Giovanni; Rossi, Maurizia. Phase singularities in complex arithmetic random waves. Electron. J. Probab. 24 (2019), paper no. 71, 45 pp. doi:10.1214/19-EJP321. https://projecteuclid.org/euclid.ejp/1561687604


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References

  • [A-T] Adler, R. J. and Taylor, J.E. (2007). Random fields and geometry. Springer-Verlag.
  • [A-L-W] Azais, J.-M., Léon, and Wschebor, M. (2011). Rice formulas and Gaussian waves. Bernoulli, 17(1), 170-193.
  • [A-W2] Azaïs, J.M. and Wschebor, M. (2009). Level Sets and Extrema of Random Processes and Fields. Wiley-Blackwell.
  • [B-M-W] D. Benatar, D. Marinucci and I. Wigman (2017). Planck-scale distribution of nodal length of arithmetic random waves. Journal d’Analyse Mathématique, in press.
  • [Be1] Berry, M.V. (1977). Regular and irregular semiclassical wavefunctions. J. Phys. A, 10(12), 2083-2091.
  • [Be2] Berry, M.V. (1978). Disruption of wavefronts: statistics of dislocations in incoherent Gaussian random waves. J. Phys. A, 11(1), 27-37.
  • [Be3] Berry, M.V. (2002). Statistics of nodal lines and points in chaotic quantum billiards: perimeter corrections, fluctuations, curvature. J. Phys. A, 35, 3025-3038.
  • [B-D] Berry, M.V., and Dennis, M. R. (2000). Phase singularities in isotropic random waves. Proc. Roy. Soc. Lond. A 456, 2059-2079.
  • [B-B] Bombieri, E. and Bourgain, J. (2015). A problem on the sum of two squares. International Math. Res. Notices, 11, 3343-3407.
  • [B-R] Bourgain, J. and Rudnick, Z. (2011). On the geometry of the nodal lines of eigenfunctions on the two-dimensional torus. Ann. Henri Poincaré, 12, 1027-1053.
  • [C] Cheng, S.Y. (1976) Eigenfunctions and nodal sets. Comment. Math. Helv., 51(1), 43-55.
  • [D-O-P] Dennis, M. R., O’Holleran, K., and Padgett, M. J. (2009). Singular Optics: Optical Vortices and Polarization Singularities. In: Progress in Optics Vol. 53., Elsevier, 293-363.
  • [D-P] Ch. Döbler and G. Peccati (2018). The Gamma Stein equation and noncentral de Jong theorems. Bernoulli 24(4B), 3384-3421.
  • [D] Dudley, R.M. (2011). Real analysis and probability. Cambridge University Press.
  • [E-H] Erdös, P., and Hall, R.R (1999). On the angular distribution of Gaussian integers with fixed norm. Discrete Math., 200(1-3), 87-94.
  • [G-H] Geman, D., and Horowitz, J. (1980). Occupation densities. Ann. Probab., 8(1), 1-67.
  • [H-W] Hardy, G.H. and Wright, E.M. (2008). An introduction to the theory of numbers (6th Edition). Oxford University Press..
  • [Ko] Khovanskiĭ, A. G. (1991). Fewnomials. American Mathematical Society.
  • [K-K-W] Krishnapur, M., Kurlberg, P., and Wigman, I. (2013).Nodal length fluctuations for arithmetic random waves. Ann. of Math. 177(2), 699-737.
  • [K-W] Kurlberg, P.; Wigman, I. (2017). On probability measures arising from lattice points on circles. Math. Ann., 367, no. 3-4, 1057-1098.
  • [L] Lee, J.M. (1997). Riemannian Manifolds. Springer-Verlag.
  • [Ma] Maffucci, R.W. (2016). Nodal intersections of random eigenfunctions against a segment on the 2-dimensional torus. Monatsh. Math., 183, no. 2, 311-328.
  • [McL] McLennan, A. (2002). The expected number of real roots of a multihomogeneous system of polynomial equations. Amer. J. Math., 124(1), 49-73.
  • [M-P-R-W] Marinucci, D., Peccati, G., Rossi, M., and Wigman, I. (2015). Non-Universality of nodal length distribution for arithmetic random waves. Geom. Funct. Anal., 26(3), 926-960.
  • [N-S] Nazarov, F. and Sodin, M. (2011). Fluctuations in random complex zeroes: Asymptotic normality revisited. Int. Math. Res. Notices, 24, 5720-5759.
  • [N-P] Nourdin, I., and Peccati, G. (2012). Normal approximations with Malliavin calculus. From Stein’s method to universality. Cambridge University Press.
  • [N] Nonnemacher, S. (2013). Anatomy of quantum chaotic eigenstates. In: B. Duplantier, S. Nonnemacher, V. Rivasseau (Eds), Chaos, Prog. Math. Phys. 66, 194-238. Birkhäuser. MR3204186
  • [N-V] Nonnemacher, S. and Voros, A. (1998). Chaotic eigenfunctions in phase space. J. Stat. Phys. 92, 431-518.
  • [N-B] Nye, J.F., and Berry, M.V. (1974). Dislocations in wave trains. Proc. R. Soc. Lond. A, 336, 165-190.
  • [O-R-W] Oravecz, F., Rudnick, Z., and Wigman, I. (2008). The Leray measure of nodal sets for random eigenfunctions on the torus. Ann. Inst. Fourier (Grenoble) 58(1), 299-335.
  • [P-R] Peccati, G. and Rossi, M. (2018). Quantitative limit theorems for local functionals of arithmetic random waves. Computation and Combinatorics in Dynamics, Stochastics and Control, The Abel Symposium, Rosendal, Norway, August 2016 13, 659-689.
  • [P-T] Peccati, G., and Taqqu, M.S. (2010). Wiener chaos: moments, cumulants and diagrams. Springer-Verlag.
  • [S-S] Shub, M., and Smale, S. (1993). Complexity of Bézout’s theorem. II. In: Volumes and Computational algebraic geometry (Nice, 1992), Progr. Math. 109, 267-285. Birkhäuser.
  • [Ro] Rossi, M. (2015). The Geometry of Spherical Random Fields. PhD thesis, University of Rome Tor Vergata. ArXiv: 1603.07575.
  • [Ro-W] Rossi, M., and Wigman, I. (2018). On the asymptotic distribution of nodal intersections of arithmetic random waves against smooth curves. Nonlinearity, 31, 4472-4516.
  • [R-W] Rudnick, Z., and Wigman, I. (2008). On the volume of nodal sets for eigenfunctions of the Laplacian on the torus. Ann. Henri Poincaré 9(1), 109-130.
  • [R-W2] Rudnick, Z., and Wigman, I. (2014). Nodal intersections for random eigenfunctions on the torus. Amer. J. of Math., 138, no. 6, 1605-1644.
  • [S-T] Sodin, M., and Tsirelson, B. (2004). Random complex zeroes, I. Asymptotic normality. Israel J. Math., 144(1), 125-149.
  • [S-Z1] Shiffman, B, and Zelditch, S. (2008). Number variance of random zeros on complex manifolds. Geom. Funct. Anal. 18(4), 1422-1475.
  • [S-Z2] Shiffman, B, and Zelditch, S. (2010). Number variance of random zeros on complex manifolds, II: smooth statistics. Pure and Applied Mathematics Quarterly, 6(4), 1145-1167.
  • [U-R] Urbina, J., and Richter, K. (2013). Random quantum states: recent developments and applications. Advances in Physics, 62, 363-452
  • [W] Wigman, I. (2010). Fluctuations of the nodal length of random spherical harmonics. Communications in Mathematical Physics, 298(3), 787-831.
  • [Ya] Yau, S.T. (1982). Survey on partial differential equations in differential geometry. Seminar on Differential Geometry, Ann. of Math. Stud. 102, 3-71. Princeton Univ. Press, Princeton.
  • [Zy] Zygmund, A. (1974). On Fourier coefficients and transforms of functions of two variables, Studia Math. 50, 189-201.