Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Quasi-independence for nodal lines

Alejandro Rivera and Hugo Vanneuville

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We prove a quasi-independence result for level sets of a planar centered stationary Gaussian field with covariance $(x,y)\mapsto\kappa(x-y)$, with only mild conditions on the regularity of $\kappa$. As a first application, we study percolation for nodal lines in the spirit of (Publ. Math. Inst. Hautes Études Sci. 126 (2017) 131–176). In the said article, Beffara and Gayet rely on Tassion’s method (Ann. Probab. 44 (2016) 3385–3398) to prove that, under some assumptions on $\kappa$, most notably that $\kappa\geq0$ and $\kappa(x)=O(|x|^{-325})$, the nodal set satisfies a box-crossing property. The decay exponent was then lowered to $16+\varepsilon$ by Beliaev and Muirhead in (Comm. Math. Phys. 359 (2018) 869–913). In the present work we lower this exponent to $4+\varepsilon$ thanks to a new approach towards quasi-independence for crossing events. This approach does not rely on quantitative discretization. Our quasi-independence result also applies to events counting nodal components and we obtain a lower concentration result for the density of nodal components around the Nazarov and Sodin constant from (Zh. Mat. Fiz. Anal. Geom. 12 (2016) 205–278).


On démontre un résultat de quasi-indépendance pour les lignes de niveau de champs gaussiens planaires stationnaires centrés de covariance $(x,y)\mapsto\kappa(x-y)$, sous de faibles conditions sur la régularité de $\kappa$. On applique d’abord ce résultat à l’étude de la percolation des lignes nodales dans l’esprit de (Publ. Math. Inst. Hautes Études Sci. 126 (2017) 131–176). Dans ledit article, Beffara et Gayet s’appuyent sur la méthode de Tassion (Ann. Probab. 44 (2016) 3385–3398) pour démontrer que sous certaines hypothèses sur $\kappa$, notamment que $\kappa\geq0$ et $\kappa(x)=O(|x|^{-325})$, l’ensemble nodal satisfait une propriété de croisement de boîtes. L’exposant de décroissance a plus tard été réduit à $16+\varepsilon$ par Beliaev et Muirhead dans (Comm. Math. Phys. 359 (2018) 869–913). Dans le présent article nous baissons cet exposant jusqu’à $4+\varepsilon$ grâce à une nouvelle approche pour la quasi-indépendance d’événements de croisement. Cette approche ne s’appuie pas sur une discrétisation quantitative. Notre résultat de quasi-indépendance s’applique aussi à des événements de comptage de composantes nodales et nous obtenons un résultat de concentration par en dessous de la densité de composantes nodales autour de la constante de Nazarov et Sodin de (Zh. Mat. Fiz. Anal. Geom. 12 (2016) 205–278).

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 3 (2019), 1679-1711.

Received: 22 December 2017
Revised: 28 June 2018
Accepted: 4 September 2018
First available in Project Euclid: 25 September 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Gaussian fields Percolation Quasi-independence Influences


Rivera, Alejandro; Vanneuville, Hugo. Quasi-independence for nodal lines. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 3, 1679--1711. doi:10.1214/18-AIHP931.

Export citation


  • [1] R. J. Adler and J. E. Taylor. Random Fields and Geometry. Springer, New York, 2007.
  • [2] D. Ahlberg, V. Tassion and A. Teixeira. Sharpness of the phase transition for continuum percolation in $\mathbb{R}^{2}$. Probab. Theory Related Fields 172 (1–2) (2017) 525–581.
  • [3] K. S. Alexander. Boundedness of level lines for two-dimensional random fields. Ann. Probab. 24 (4) (1996) 1653–1674.
  • [4] J.-M. Azaïs and M. Wschebor. Level Sets and Extrema of Random Processes and Fields. Wiley, Hoboken, NJ, 2009.
  • [5] D. Basu and A. Sapozhnikov. Crossing probabilities for critical Bernoulli percolation on slabs. Preprint, 2015. Available at arXiv:1512.05178.
  • [6] V. Beffara and D. Gayet. Percolation of random nodal lines. Publ. Math. Inst. Hautes Études Sci. 126 (2017) 131–176.
  • [7] V. Beffara and D. Gayet. Percolation without FKG. Preprint, 2017. Available at arXiv:1710.10644.
  • [8] D. Beliaev and S. Muirhead. Discretisation schemes for level sets of planar Gaussian fields. Comm. Math. Phys. 359 (3) (2018) 869–913.
  • [9] D. Beliaev, S. Muirhead and I. Wigman. Russo–Seymour–Welsh estimates for the Kostlan ensemble of random polynomials. Preprint, 2017. Available at arXiv:1709.08961.
  • [10] E. Bogomolny and C. Schmit. Random wavefunctions and percolation. J. Phys. A: Math. Theor. 40 (47) (2007) 14033–14043.
  • [11] B. Bollobás and O. Riordan. The critical probability for random Voronoi percolation in the plane is $1/2$. Probab. Theory Related Fields 136 (3) (2006) 417–468.
  • [12] B. Bollobás and O. Riordan. Percolation. Cambridge University Press, Cambridge, 2006.
  • [13] E. W. Cheney and W. A. Light. A Course in Approximation Theory. Graduate Studies in Mathematics 101. American Mathematical Society, Providence, RI, 2009.
  • [14] H. Duminil-Copin, C. Hongler and P. Nolin. Connection probabilities and RSW-type bounds for the two-dimensional FK Ising model. Comm. Pure Appl. Math. 64 (9) (2011) 1165–1198.
  • [15] H. Duminil-Copin, V. Tassion and A. Teixeira. The box-crossing property for critical two-dimensional oriented percolation. Probab. Theory Related Fields 171 (3–4) (2018) 685–708.
  • [16] D. Gayet and J.-Y. Welschinger. Exponential rarefaction of real curves with many components. Publ. Math. Inst. Hautes Études Sci. 113 (2011) 69–96.
  • [17] G. Grimmett. Probability on Graphs. Random Processes on Graphs and Lattices. Cambridge University Press, Cambridge, 2010.
  • [18] G. R. Grimmett. Percolation. Grundlehren der mathematischen Wissenschaften. Springer, Berlin, 1999.
  • [19] T. E. Harris. A lower bound for the critical probability in a certain percolation process. Proc. Camb. Philos. Soc. 56 (1960) 13–20.
  • [20] H. Kesten. The critical probability of bond percolation on the square lattice equals $\frac{1}{2}$. Comm. Math. Phys. 74 (1) (1980) 41–59.
  • [21] T. Letendre. Variance of the volume of random real algebraic submanifolds. Preprint, 2016. Available at arXiv:1608.05658.
  • [22] S. A. Molchanov and A. K. Stepanov. Percolation in random fields. I. Theoret. Math. Phys. 55 (2) (1983) 478–484.
  • [23] S. A. Molchanov and A. K. Stepanov. Percolation in random fields. II. Theoret. Math. Phys. 55 (3) (1983) 592–599.
  • [24] S. A. Molchanov and A. K. Stepanov. Percolation in random fields. III. Theoret. Math. Phys. 67 (2) (1986) 434–439.
  • [25] F. Nazarov and M. Sodin. On the number of nodal domains of random spherical harmonics. Amer. J. Math. 131 (5) (2009) 1337–1357.
  • [26] F. Nazarov and M. Sodin. Fluctuations in random complex zeroes: Asymptotic normality revisited. Int. Math. Res. Not. IMRN 24 (2011) 5720–5759.
  • [27] 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 (3) (2016) 205–278.
  • [28] F. Nazarov, M. Sodin and A. Volberg. Transportation to random zeroes by the gradient flow. Geom. Funct. Anal. 17 (3) (2007) 887–935.
  • [29] F. Nazarov, M. Sodin and A. Volberg. The Jancovici–Lebowitz–Manificat law for large fluctuations of random complex zeroes. Comm. Math. Phys. 284 (3) (2008) 833–865.
  • [30] C. Newman, V. Tassion and W. Wu. Critical percolation and the minimal spanning tree in slabs. Comm. Pure Appl. Math. 70 (11) (2017) 2084–2120.
  • [31] V. I. Piterbarg. Asymptotic Methods in the Theory of Gaussian Processes and Fields. American Mathematical Society, Providence, RI, 1996. Transl. from the Russian by V. V. Piterbarg. Transl. ed. by Simeon Ivanov.
  • [32] L. D. Pitt. Positively correlated normal variables are associated. Ann. Probab. 10 (1982) 496–499.
  • [33] A. Rivera and H. Vanneuville. The critical threshold for Bargmann–Fock percolation. Preprint, 2017. Available at arXiv:1711.05012.
  • [34] L. Russo. A note on percolation. Probab. Theory Related Fields 43 (1) (1978) 39–48.
  • [35] P. D. Seymour and D. J. A. Welsh. Percolation probabilities on the square lattice. Ann. Discrete Math. 3 (1978) 227–245.
  • [36] D. Slepian. The one-sided barrier problem for Gaussian noise. Bell Labs Tech. J. 41 (2) (1962) 463–501.
  • [37] V. Tassion. Crossing probabilities for Voronoi percolation. Ann. Probab. 44 (5) (2016) 3385–3398.