Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Fluctuations of the Euler-Poincaré characteristic for random spherical harmonics


Authors: V. Cammarota, D. Marinucci and I. Wigman
Journal: Proc. Amer. Math. Soc. 144 (2016), 4759-4775
MSC (2010): Primary 33C55, 42C10, 60D05, 60B10, 60G60
DOI: https://doi.org/10.1090/proc/13299
Published electronically: August 1, 2016
MathSciNet review: 3544528
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this short note, we build upon recent results from our earlier paper to present a precise expression for the asymptotic variance of the Euler-Poincaré characteristic for the excursion sets of Gaussian eigenfunctions on $ \mathcal {S}^2$; this result can be written as a second-order Gaussian kinematic formula for the excursion sets of random spherical harmonics. The covariance between the Euler-Poincaré characteristics for different level sets is shown to be fully degenerate; it is also proved that the variance for the zero level excursion sets is asymptotically of smaller order.


References [Enhancements On Off] (What's this?)

  • [1] R. J. Adler, K. Bartz, S. C. Kou, and A. Monod, Estimating thresholding levels for random fields via Euler characteristics. Preprint, http://webee.technion.ac.il/people/adler/LKC-AAS.pdf
  • [2] Robert J. Adler and Jonathan E. Taylor, Random fields and geometry, Springer Monographs in Mathematics, Springer, New York, 2007. MR 2319516
  • [3] Robert J. Adler and Jonathan E. Taylor, Topological complexity of smooth random functions, Lectures from the 39th Probability Summer School held in Saint-Flour, 2009; École d'Été de Probabilités de Saint-Flour. [Saint-Flour Probability Summer School], Lecture Notes in Mathematics, vol. 2019, Springer, Heidelberg, 2011. MR 2768175
  • [4] George E. Andrews, Richard Askey, and Ranjan Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR 1688958
  • [5] Jean-Marc Azaïs and Mario Wschebor, Level sets and extrema of random processes and fields, John Wiley & Sons, Inc., Hoboken, NJ, 2009. MR 2478201
  • [6] M. V. Berry, Regular and irregular semiclassical wavefunctions, J. Phys. A 10 (1977), no. 12, 2083-2091. MR 0489542
  • [7] V. Cammarota, D. Marinucci, and I. Wigman, On the distribution of the critical values of random spherical harmonics, The Journal of Geometric Analysis. To appear. arXiv:1409.1364 [math-ph].
  • [8] V. Cammarota, Y. Fantaye, D. Marinucci, and I. Wigman, In preparation.
  • [9] V. Cammarota and I. Wigman, Fluctuations of the total number of critical points of random spherical harmonics. arXiv:1510.00339 [math-ph].
  • [10] Isaac Chavel, Riemannian geometry, A modern introduction, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 98, Cambridge University Press, Cambridge, 2006. MR 2229062
  • [11] Dan Cheng and Yimin Xiao, Excursion probability of Gaussian random fields on sphere, Bernoulli 22 (2016), no. 2, 1113-1130. MR 3449810, https://doi.org/10.3150/14-BEJ688
  • [12] Dan Cheng and Yimin Xiao, The mean Euler characteristic and excursion probability of Gaussian random fields with stationary increments, Ann. Appl. Probab. 26 (2016), no. 2, 722-759. MR 3476623, https://doi.org/10.1214/15-AAP1101
  • [13] Domenico Marinucci and Giovanni Peccati, Random fields on the sphere, Representation, limit theorems and cosmological applications, London Mathematical Society Lecture Note Series, vol. 389, Cambridge University Press, Cambridge, 2011. MR 2840154
  • [14] Domenico Marinucci and Sreekar Vadlamani, High-frequency asymptotics for Lipschitz-Killing curvatures of excursion sets on the sphere, Ann. Appl. Probab. 26 (2016), no. 1, 462-506. MR 3449324, https://doi.org/10.1214/15-AAP1097
  • [15] Domenico Marinucci and Igor Wigman, On the area of excursion sets of spherical Gaussian eigenfunctions, J. Math. Phys. 52 (2011), no. 9, 093301, 21. MR 2867816, https://doi.org/10.1063/1.3624746
  • [16] Domenico Marinucci and Maurizia Rossi, Stein-Malliavin approximations for nonlinear functionals of random eigenfunctions on $ \mathbb{S}^d$, J. Funct. Anal. 268 (2015), no. 8, 2379-2420. MR 3318653, https://doi.org/10.1016/j.jfa.2015.02.004
  • [17] Z. Rudnick and I. Wigman, Nodal intersections for random eigenfunctions on the torus, Amer. J. of Math.. To appear. arXiv:1402.3621 [math-ph].
  • [18] Z. Rudnick, I. Wigman, and N. Yesha, Nodal intersections for random waves on the 3-dimensional torus. arXiv:1501.07410 [math-ph].
  • [19] Jonathan E. Taylor, A Gaussian kinematic formula, Ann. Probab. 34 (2006), no. 1, 122-158. MR 2206344, https://doi.org/10.1214/009117905000000594
  • [20] Igor Wigman, Fluctuations of the nodal length of random spherical harmonics, Comm. Math. Phys. 298 (2010), no. 3, 787-831. MR 2670928, https://doi.org/10.1007/s00220-010-1078-8
  • [21] Igor Wigman, On the nodal lines of random and deterministic Laplace eigenfunctions, Spectral geometry, Proc. Sympos. Pure Math., vol. 84, Amer. Math. Soc., Providence, RI, 2012, pp. 285-297. MR 2985322, https://doi.org/10.1090/pspum/084/1362

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 33C55, 42C10, 60D05, 60B10, 60G60

Retrieve articles in all journals with MSC (2010): 33C55, 42C10, 60D05, 60B10, 60G60


Additional Information

V. Cammarota
Affiliation: Department of Mathematics, Università degli Studi di Roma Tor Vergata, 00133 Rome, Italy
Email: cammarot@mat.uniroma2.it

D. Marinucci
Affiliation: Department of Mathematics, Università degli Studi di Roma Tor Vergata, 00133 Rome, Italy
Email: marinucc@mat.uniroma2.it

I. Wigman
Affiliation: Department of Mathematics, King’s College London, Strand, London WC2 2LS, England
Email: igor.wigman@kcl.ac.uk

DOI: https://doi.org/10.1090/proc/13299
Received by editor(s): April 8, 2015
Received by editor(s) in revised form: December 22, 2015
Published electronically: August 1, 2016
Additional Notes: The first and second author’s research was supported by ERC grant No. 277742
The third author’s research was supported by ERC grant No. 335141
Communicated by: Mark M. Meerschaert
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society